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Kakeya-type sets, lacunarity, and directional maximal operators in Euclidean space Kroc, Edward 2015

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Kakeya-type Sets, Lacunarity, and DirectionalMaximal Operators in Euclidean SpacebyEdward KrocA thesis submitted in partial fulfilment of the requirements forthe degree ofDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)The University of British Columbia(Vancouver)April 2015c© Edward Kroc, 2015AbstractGiven a Cantor-type subset Ω of a smooth curve in Rd+1, we construct randomexamples of Euclidean sets that contain unit line segments with directions from Ω andenjoy analytical features similar to those of traditional Kakeya sets of infinitesimalLebesgue measure. We also develop a notion of finite order lacunarity for directionsets in Rd+1, and use it to extend our construction to direction sets Ω that aresublacunary according to this definition. This generalizes to higher dimensions a pairof planar results due to Bateman and Katz [4], [3]. In particular, the existence ofsuch sets implies that the directional maximal operator associated with the directionset Ω is unbounded on Lp(Rd+1) for all 1 ≤ p <∞.iiPrefaceMuch of the proceeding document is adapted from two research papers authored bymyself and Malabika Pramanik, currently unpublished. These materials are usedwith permission. Chapters 6 through 11 form the main content of [31], Kakeya-typesets over Cantor sets of directions in Rd+1, while Chapters 2, 3.7, and 12 through19 are adapted from [32], Lacunarity, Kakeya-type sets and directional maximal op-erators. The first of these two manuscripts has recently been conditionally acceptedfor publication in the Journal of Fourier Analysis and Applications.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Notations, conventions, and structure of the document . . . . . . . . 61.3 Early history of Kakeya sets . . . . . . . . . . . . . . . . . . . . . . . 101.4 Background: maximal averages over lines with prescribed directions . 171.5 Kakeya-type sets and the property of stickiness . . . . . . . . . . . . 212 Finite order lacunarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1 Lacunarity on the real line . . . . . . . . . . . . . . . . . . . . . . . . 242.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.2 Non-closure of finite order lacunarity under algebraic sums . . 282.2 Finite order lacunarity in general dimensions . . . . . . . . . . . . . . 292.2.1 Examples of admissible lacunary and sublacunary sets in Rd . 322.3 Finite order lacunarity for direction sets . . . . . . . . . . . . . . . . 352.3.1 Examples of admissible lacunary and sublacunary direction sets 36iv3 Rooted, labelled trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1 The terminology of trees . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Encoding bounded subsets of the unit interval by trees . . . . . . . . 433.3 Encoding higher dimensional bounded subsets of Euclidean space bytrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Stickiness as a kind of mapping between trees . . . . . . . . . . . . . 463.5 The tree structure of Cantor-type and sublacunary sets . . . . . . . . 483.6 The splitting number of a tree . . . . . . . . . . . . . . . . . . . . . . 503.6.1 Preliminary facts about splitting numbers . . . . . . . . . . . 523.6.2 A reformulation of Theorem 1.3 . . . . . . . . . . . . . . . . . 533.7 Lacunarity on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 Electrical circuits and percolation on trees . . . . . . . . . . . . . . 604.1 The percolation process associated to a tree . . . . . . . . . . . . . . 604.2 Trees as electrical networks . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Estimating the survival probability after percolation . . . . . . . . . . 655 Kakeya-type sets over a Cantor set of directions in the plane . . . 685.1 The result of Bateman and Katz . . . . . . . . . . . . . . . . . . . . . 685.1.1 Proof of inequality (5.4) . . . . . . . . . . . . . . . . . . . . . 705.1.2 Proof of inequality (5.5) . . . . . . . . . . . . . . . . . . . . . 745.2 Points of distinction between the construction of Kakeya-type sets overCantor directions in the plane and over arbitrary sublacunary sets inany dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 Setup of construction of Kakeya-type sets in Rd+1 over a Cantorset of directions: a reformulation of Theorem 1.2 . . . . . . . . . . 807 Families of intersecting tubes . . . . . . . . . . . . . . . . . . . . . . . 858 The random mechanism and sticky collections of tubes in Rd+1over a Cantor set of directions . . . . . . . . . . . . . . . . . . . . . . 91v8.1 Slope assignment algorithm . . . . . . . . . . . . . . . . . . . . . . . 938.2 Construction of Kakeya-type sets revisited . . . . . . . . . . . . . . . 969 Slope probabilities and root configurations, Cantor case . . . . . . 979.1 Four point root configurations . . . . . . . . . . . . . . . . . . . . . . 1009.2 Three point root configurations . . . . . . . . . . . . . . . . . . . . . 10310 Proposition 8.4: proof of the lower bound (6.6) . . . . . . . . . . . . 10510.1 Proof of Proposition 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . 10710.2 Proof of Proposition 10.2 . . . . . . . . . . . . . . . . . . . . . . . . . 10910.3 Expected intersection counts . . . . . . . . . . . . . . . . . . . . . . . 11311 Proposition 8.4: proof of the upper bound (6.7) . . . . . . . . . . . 12212 Construction of Kakeya-type sets in Rd+1 over an arbitrary sub-lacunary set of directions . . . . . . . . . . . . . . . . . . . . . . . . . . 12812.1 Pruning of the slope tree . . . . . . . . . . . . . . . . . . . . . . . . . 12912.2 Splitting and basic slope cubes . . . . . . . . . . . . . . . . . . . . . . 13512.3 Binary representation of ΩN . . . . . . . . . . . . . . . . . . . . . . . 13813 Families of intersecting tubes, revisited . . . . . . . . . . . . . . . . . 14013.1 Tubes and a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14213.2 Weakly sticky maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14814 Random construction of Kakeya-type sets . . . . . . . . . . . . . . . 15214.1 Features of the construction . . . . . . . . . . . . . . . . . . . . . . . 15514.2 Theorem 1.3 revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 15715 Proof of the upper bound (14.10) . . . . . . . . . . . . . . . . . . . . . 15916 Probability estimates for slope assignments . . . . . . . . . . . . . . 16316.1 A general rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16316.2 Root configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16616.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167vi16.4 The case of two roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 16816.5 The case of three roots . . . . . . . . . . . . . . . . . . . . . . . . . . 16916.6 The case of four roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 17317 Tube counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18317.1 Collections of two intersecting tubes . . . . . . . . . . . . . . . . . . . 18317.2 Counting slope tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . 18817.3 Collections of four tubes with at least two pairwise intersections . . . 19017.3.1 Four roots of type 1 . . . . . . . . . . . . . . . . . . . . . . . 19017.3.2 Four roots of type 2 . . . . . . . . . . . . . . . . . . . . . . . 19117.3.3 Four roots of type 3 . . . . . . . . . . . . . . . . . . . . . . . 19417.4 Collections of three tubes with at least two pairwise intersections . . 19818 Sums over root and slope vertices . . . . . . . . . . . . . . . . . . . . 20019 Proof of the lower bound (14.9) . . . . . . . . . . . . . . . . . . . . . . 20519.1 Proof of Proposition 19.1 . . . . . . . . . . . . . . . . . . . . . . . . . 20619.2 Proof of Proposition 19.2 . . . . . . . . . . . . . . . . . . . . . . . . . 20819.2.1 Expected value of S41 . . . . . . . . . . . . . . . . . . . . . . 21019.2.2 Expected value of S42 . . . . . . . . . . . . . . . . . . . . . . 21319.2.3 Expected value of S43 . . . . . . . . . . . . . . . . . . . . . . 21920 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22420.1 Maximal functions over other collections of sticky objects . . . . . . . 22420.2 A characterization of the Lp(Rd+1)-boundedness of directional maxi-mal operators over an arbitrary set of directions . . . . . . . . . . . . 22720.2.1 Boundedness of directional maximal operators, sketch . . . . . 22820.2.2 Boundedness of directional maximal operators, a detailed ex-ample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237viiList of Figures1.1 A direction set in the plane . . . . . . . . . . . . . . . . . . . . . . . 61.2 Flowchart of thesis chapters . . . . . . . . . . . . . . . . . . . . . . . 71.3 Flowchart of thesis chapters, proof of the main theorems . . . . . . . 81.4 Besicovitch construction of a zero measure Kakeya set . . . . . . . . . 111.5 Besicovitch’s solution to the Kakeya needle problem . . . . . . . . . . 131.6 A sticky collection of tubes . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Cantor set tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Example trees encoding simple Euclidean sets . . . . . . . . . . . . . 513.3 A figure explaining inequality (3.15) when M = 2 and nk = k. . . . . 594.1 Diagram of proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . 636.1 Two tubes in the Kakeya-type set . . . . . . . . . . . . . . . . . . . . 827.1 Triangle defined by the intersection of two tubes . . . . . . . . . . . . 867.2 The cone generated by a set of Cantor directions . . . . . . . . . . . . 889.1 A sticky assignment between trees . . . . . . . . . . . . . . . . . . . . 989.2 Intersections of tubes, four point configurations of type 1 . . . . . . . 1019.3 Intersections of tubes, four point configurations of type 2 . . . . . . . 1039.4 Intersections of tubes, three point configurations . . . . . . . . . . . . 10410.1 Diagram of all root cubes restricted to lie within a prescribed distancefrom the boundary of a child of some tree vertex . . . . . . . . . . . . 114viii12.1 Euclidean separation in the pruning mechanism of the slope tree . . . 13312.2 Illustration of a pruned tree at the second step of pruning . . . . . . 13512.3 Two basic slope cubes . . . . . . . . . . . . . . . . . . . . . . . . . . 13713.1 The pull-back mechanism used to define the compressed root tree . . 14514.1 Diagram of a typical weakly sticky slope assignment . . . . . . . . . . 15416.1 Identification of basic slope cubes . . . . . . . . . . . . . . . . . . . . 16816.2 Intersections of tubes, three point configurations . . . . . . . . . . . . 17016.3 Intersections of tubes, four point configurations . . . . . . . . . . . . 17516.4 Root configurations not preserved under sticky mappings . . . . . . . 18217.1 Illustration of the proof of Lemma 17.1 . . . . . . . . . . . . . . . . . 18517.2 Illustration of the proof of Lemma 17.2 . . . . . . . . . . . . . . . . . 18717.3 Illustration of the spatial restriction of a root cube paired with a cer-tain structured partner root cube . . . . . . . . . . . . . . . . . . . . 19419.1 Slope configurations for four roots of type 3 . . . . . . . . . . . . . . 223ixAcknowledgementsAbove all others, I would like to thank my supervisor Malabika Pramanik. This workwould not have been remotely possible without her unwavering support and tirelesscommitment. Her advice, her guidance, and her enthusiasm have been invaluable.I am grateful for the financial support she has granted me throughout my graduatestudies at UBC, support that has allowed me to fulfill a personal dream of studyingmathematics professionally, while simultaneously pursuing my myriad other interestsand ambitions. Malabika has generously supported me in all my scientific and profes-sional endeavours, regardless of their bearing on my status as a mathematician. Shehas guided me through the sometimes existential perils of life in academia, providingtruly life changing direction and support. She has been there for me at each andevery step of my academic career over the past six years, always happy to help inany way possible. I will be forever grateful for her kindness, her assistance, and herwisdom. I consider myself immeasurably lucky to have had the chance to work withand learn from her. She is my mentor, my colleague, and my friend.I would also like to thank Paul Gustafson, Derek Bingham, Tim McMurry, andAhmed Zayed. They have each provided unique and invaluable support at variousstages in my career as a graduate student. They are significantly responsible forshaping the type of scientist I am today, and have provided much of the model forthe type I would like to become. Paul and Derek in particular have helped guideme through some of the roughest parts of this process, providing abundant encour-agement and inspiration when these were most needed. I will be forever indebted tothem for their generosity.I want to thank the members of my committee, Izabella  Laba, Ozgur Yilmaz,xand Akos Magyar, for their advice, their support, and their time.Beyond vocational or academic matters, I want to thank the Mathematics de-partment administration and staff. No other department on campus has such excep-tional staff who work so hard at making their department function smoothly. Theyare consummate professionals and wonderful people. In particular, I want to thankLee Yupitun for her friendship and for all her help navigating the graduate programat UBC.I would be extremely remiss not to acknowledge the unshakable support of mypartner, my parents, and my friends throughout the drafting of this document, aswell as throughout my entire graduate student career. I would never have pursuedthe dream of a Ph.D., especially not in such a wonderful place as Vancouver, BritishColumbia, without the encouragement of my partner, Marybeth Welty. She hasalways kept me afloat in the worst of storms, and has made all the best parts of thisprocess orders of magnitude better. My parents have helped me financially, mentally,and emotionally throughout this process, and have always been happy to provide atemporary but comfortable escape from it. For that, I am especially grateful. Mybest and oldest friends Fred Drueck and Darin Hatcher remain incomparable in theirability to keep me grounded and thankful for what I have.Finally, I want to thank my graduate colleagues in and outside the Mathematicsdepartment at UBC, in particular those who spend most of their time boxed up inthe Auditorium Annex A offices. The open and easy atmosphere they create makesa day at the office feel like an escape amongst friends.xiChapter 1IntroductionThe main focus of this document is a study of Kakeya-type sets in Euclidean space ofgeneral dimensions. Such sets have much in common with the now classic Besicovitchconstruction of Kakeya sets with zero Lebesgue measure [6]. Interest in the analyticalproperties of these sets arises in both the traditional study of Kakeya sets in Euclideanspace, as well as from a well-known connection between the existence of such setsand the boundedness properties of various maximal operators over Lp-space, notablythe generalized one-dimensional Hardy-Littlewood maximal operator; see (1.8) and(1.2).Definition 1.1. Fix a set of directions Ω ⊆ Sd. We say a cylindrical tube is orientedin direction ω ∈ Ω if the principal axis of the cylinder is parallel to ω. If for somefixed constant A0 ≥ 1 and any choice of integer N ≥ 1, there exist- a number 0 < δN  1, δN ↘ 0 as N ↗∞, and- a collection of tubes {P (N)t } with orientations in Ω, length at least 1 and cross-sectional radius at most δNobeyinglimN→∞|E∗N(A0)||EN |= ∞, with EN :=⋃tP (N)t , E∗N(A0) :=⋃tA0P (N)t , (1.1)1then we say that Ω admits Kakeya-type sets. Here, | · | denotes (d+ 1)-dimensionalLebesgue measure, and A0P (N)t denotes the tube with the same centre, orientationand cross-sectional radius as P (N)t but A0 times its length. The tubes that constituteEN may have variable dimensions subject to the restrictions mentioned above. Werefer to {EN : N ≥ 1} as sets of Kakeya type.We will explore the motivations and the history behind this definition in Sec-tion 1.5, but for now it suffices to reiterate that sets of Kakeya type, according tothis Definition 1.1, share a critical feature with known constructions of zero measureKakeya sets. This feature is intimately related to the notion of stickiness, arisingfrom the study of classical Kakeya sets (see e.g. [46], [27]). We will describe theseconnections and discuss their history in Sections 1.3–1.5, but roughly speaking stick-iness means the following: the tubes {P (N)t } have considerable overlap, while theirdilates {A0P (N)t } are comparatively disjoint.1.1 Summary of resultsIn brief, the new results contained in this document generalize the work of Batemanand Katz [4] and of Bateman [3] in the plane to an arbitrary number of dimensions.Our main body of work is devoted to establishing the existence of Kakeya-type sets,as defined in Definition 1.1, for certain direction sets Ω ⊆ Sd. These results willhave immediate consequences for the Lp-boundedness of two often studied maximaloperators (see (1.2) and (1.3) below). Indeed, as we will explain in Section 1.4, ifΩ ⊂ Sd admits Kakeya-type sets, then both of these operators are unbounded onLp(Rd+1) for all p ∈ [1,∞).Given a set of directions Ω, the directional maximal operator DΩ is defined byDΩf(x) := supω∈Ωsuph>012h∫ h−h|f(x+ ωt)|dt, (1.2)where f : Rd+1 → C is a function that is locally integrable along lines. Also, forany locally integrable function f on Rd+1, we consider the Kakeya-Nikodym maximal2operator MΩ defined byMΩf(x) := supω∈ΩsupP3xP‖ω1|P |∫P|f(y)|dy, (1.3)where the inner supremum is taken over all cylindrical tubes P containing the pointx, oriented in the direction ω. The tubes are taken to be of arbitrary length l andhave circular cross-section of arbitrary radius r, with r ≤ l.Bateman and Katz [4] show that Ω ⊂ S1 admits Kakeya-type sets when Ω ={(cos θ, sin θ) : θ ∈ C1/3} and C1/3 is the ordinary middle-third Cantor set on [0, 1].We extend this result to a general (d + 1)-dimensional setting, using the followingnotion of a Cantor set of directions in (d+ 1) dimensions.Fix some integer M ≥ 3. Construct an arbitrary Cantor-type subset of [0, 1) asfollows.• Partition [0, 1] into M subintervals of the form [a, b], all of equal length M−1.Among these M subintervals, choose any two that are not adjacent (i.e., donot share a common endpoint); define C[1]M to be the union of these chosensubintervals, called first stage basic intervals.• Partition each first stage basic interval into M further (second stage) subin-tervals of the form [a, b], all of equal length M−2. Choose two non-adjacentsecond stage subintervals from each first stage basic one, and define C[2]M to bethe union of the four chosen second stage (basic) intervals.• Repeat this procedure ad infinitum, obtaining a nested, non-increasing se-quence of sets. Denote the limiting set by CM :CM =∞⋂k=1C[k]M .We call CM a generalized Cantor-type set (with base M).While conventional uniform Cantor sets, such as the Cantor middle-third set, arespecial cases of generalized Cantor-type sets, the latter may not in general look3like the former. In particular, sets of the form CM need not be self-similar. It iswell-known (see [16, Chapter 4]) that such sets have Hausdorff dimension at mostlog 2/ logM . By choosing M large enough, we can thus construct generalized Cantor-type sets of arbitrarily small dimension.In Chapters 8–11, we prove the following [31].Theorem 1.2. (Kroc, Pramanik) Let CM ⊂ [0, 1] be a generalized Cantor-type setdescribed above. Let γ : [0, 1] → {1} × [−1, 1]d be an injective map that satisfies abi-Lipschitz condition∀ x, y, c|x− y| ≤ |γ(x) − γ(y)| ≤ C|x− y|, (1.4)for some absolute constants 0 < c < 1 < C <∞. Set Ω = {γ(t) : t ∈ CM}. Then(i) the set Ω admits Kakeya-type sets;(ii) the operators DΩ and MΩ are unbounded on Lp(Rd+1) for all 1 ≤ p <∞.Part (ii) of the theorem follows directly from part (i) as we will see shortly inSection 1.3. The condition in Theorem 1.2 that γ satisfies a bi-Lipschitz condition canbe weakened, but it will help in establishing some relevant geometry. It is instructiveto envision γ as a smooth curve on the plane {x1 = 1}, and we recommend the readerdoes this to aid in visualization. Our underlying direction set of interest Ω = γ(CM)is essentially a Cantor-type subset of this curve.After working through the details of Theorem 1.2, we generalize many of theideas considerably to establish the following [32].Theorem 1.3. (Kroc, Pramanik) Let d ≥ 1. If the direction set Ω ⊆ Rd+1 issublacunary in the sense of Definition 2.7, then(i) Ω admits Kakeya-type sets;(ii) the operators DΩ and MΩ are unbounded on Lp(Rd+1) for all 1 ≤ p <∞.4Again, part (ii) of the theorem follows from part (i) by the same mechanism to bedescribed in Section 1.3. Precise definitions of lacunarity and sublacunarity needed inthis document are deferred to Chapter 2, but the general idea is easy to describe. Inone dimension, a relatively compact set {ai} is lacunary of order 1 if there is a pointa ∈ R and some positive λ < 1 such that |ai+1 − a| ≤ λ|ai − a| for all i. Such a sethas traditionally been referred to as a lacunary sequence with lacunarity constant(at most) λ. A lacunary set of order 2 consists of a single (first-level) lacunarysequence {ai}, along with a collection of disjoint (second-level) lacunary sequences;a second-level sequence is squeezed between two adjacent elements of {ai}. Thelacunarity constants of all sequences are uniformly bounded by some positive λ < 1.See Figure 1.1 for an illustration of a lacunary set of directions in the plane of order2.Roughly speaking, a set on the real line is lacunary of finite order if there is adecomposition of the real line by points of a lacunary sequence such that the restric-tion of the set to each of the resulting subintervals is lacunary of lower order. Alllacunarity constants implicit in the definition are assumed to be uniformly boundedaway from unity. A set is then said to be sublacunary if it does not admit a finitecovering by lacunary sets of finite order. A Cantor-like set is a particular kind ofsublacunary set according to these definitions.As noted before, we will devise a formal definition of sublacunarity in Chapter 2.However, for analytical purposes, we will see that it is in fact more convenient torecast the lacunarity (or lack therof) of a set in terms of its encoded tree structure.This idea was first formally noted in the work of Bateman and Katz [4], and sub-sequently played a role in the later work of Bateman [3]. We will discuss the treestructure of sets in Chapter 3, and see that the inherent tree structure of a sublacu-nary (in particular, Cantor-like) subset of Euclidean space is, in a quantifiable sense,as full as that of any nonzero Lebesgue measure Euclidean set; see Propositions 3.2and 3.6. In the meantime, it is recommended that the reader keep in mind thatsublacunary sets are, in a rather fundamental sense to be explored in Section 3.6,fully generalized Cantor-like subsets of Euclidean space.5λ3λ2λλ + γk1λ + γk1+1λ + γk1+2λ2 + γk2λ2 + γk2+1λ2 + γk2+2λ3 + γk3λ3 + γk3+1λ3 + γk3+2Figure 1.1: A direction set in the plane, represented as a collection of unit vectors,with parameters 0 < γ < λ < 1/2. The set of angles made by these vectors with thepositive horizontal axis is {(λj + γk) : k ≥ j}, which is lacunary of order 2.1.2 Notations, conventions, and structure of thedocumentUnless otherwise stated, for A ⊆ Rd, with d minimal, the notation |A| will always beused to denote the d-dimensional Lebesgue measure of A. For a countable, possiblyinfinite set Q, the notation #(Q) will denote cardinality of the set Q. If XN andYN are quantities depending on N , then we will write XN . YN to mean that thereexists a constant C independent of N such that XN ≤ CYN . If both XN . YN andYN . XN hold, then we will write XN ∼ YN .As in Definition 1.1, if P is a tube with some arbitrary centre, orientation, cross-sectional radius, and length l, and if C0 is a positive constant, then C0P will denotethe tube with the same centre, orientation, and cross-sectional radius as P , but withlength C0l.6This document is divided into twenty main chapters, and each chapter falls inone of four expository groups, each with a single, broad goal. Chapters 1–5 setdefinitions and discuss the necessary background, Chapters 6–11 form the proof ofTheorem 1.2, and Chapters 12–19 deal with the proof of Theorem 1.3; the finalChapter 20 discuss potential future work that could arise from this document. Theflow charts below diagram the logical ordering of the individual chapters containedwithin this document.§1§3§2§4§3.6§5ABTheorem 1.2Theorem 1.3Figure 1.2: Diagram illustrating the approximate dependence structure betweenchapters in this paper, with respect to the proofs of Theorems 1.2 and 1.3. Thechapter structure of groups A and B are detailed in the two maps below.Chapter 1 is introductory and discusses the history of the problems consideredin this document, as well as the motivation for their study. Chapter 2 discusses thenotion of lacunarity. We lay out precise definitions, check for consistency with theestablished literature, and are able to properly state Theorem 1.3 with respect tothese definitions. This chapter does not directly pertain to the proof of Theorem 1.2and can be skipped until the reader begins the chapters of group B.Chapter 3 introduces the critical idea of a tree encoding a set in Euclidean space,and Section 3.7 discusses the idea of lacunarity on trees (not needed until the chaptersof group B). The so-called splitting number of a tree, as defined in [3], is then shownto be the critical concept that allows us to recast the notion of (admissible) finiteorder lacunarity of a set into an equivalent and more tractable form for the purposesof our proof. Chapter 4 reviews the relevant literature pertaining to percolation ontrees. Chapter 5 describes the known results about Kakeya-type sets in the plane,7A§6§7§8§11§9§10Theorem 1.2B§12§13§14§18§17§16§15§19Theorem 1.3Figure 1.3: Diagram illustrating the approximate dependence structure betweenchapters in groups A and B of Figure 1.2. The chapters in group A detail the proofof Theorem 1.2, while the chapters in group B constitute the proof of Theorem 1.3.Dotted arrows indicate a dependence in terms of definitions and notation only.and presents the proof of Bateman and Katz [4] in detail. The material in thischapter is not directly required in subsequent chapters and can be skipped if desired.Chapters 6–11 form the proof of Theorem 1.2. Chapter 6 sets up the constructionof Kakeya-type sets over a Cantor set of directions, and presents a probabilisticversion of our main theorems. Chapter 7 explores the relevant geometry of theintersection of two tubes in Euclidean space, recording facts for later use. Chapter 8combines results of the previous two chapters to explicitly describe the probabilisticmechanism in play. We also reformulate Theorem 1.2 in terms of quantitative upperand lower bounds on the sizes of a typical Kakeya-type set EN and its principaldilate E∗N(A0) as described in Definition 1.1 (see Proposition 8.4). From here, the8remaining chapters in group A split into more or less two disjoint expositions, eachone charged with establishing one of these two probabilistic and quantitative bounds.Chapters 9 and 10 combine to establish the quantitative lower bound. Chapter 9in particular explores the analytical implications of the structure imposed on theposition and slope trees of a collection of two, three, or four δ-tubes, certain pairs ofwhich are required to intersect at a given location in space. In Chapter 11 we provethe quantitative upper bound using an argument similar to [4].Chapter 12 begins the program of proving Theorem 1.3; i.e., of constructingKakeya-type sets over sublacunary direction sets. The structure of the chapters ofgroup B mirrors that of group A just described.We use the language of trees developed in Chapter 3 to extract a convenient sub-set of an arbitrary sublacunary direction set, denoted by ΩN . Chapter 13 expands onthe geometry of the intersection of two tubes as initiated in Chapter 7 and the im-plications of this geometry for the structure of trees encoding the sets of orientationsand positions of a given collection of thin δ-tubes.Chapter 14 combines results from the previous two chapters to describe the ac-tual mechanism we use to assign slopes in ΩN to δ-tubes affixed to a prescribed setof points in Euclidean space. Again, we reformulate Theorem 1.3 in terms of quanti-tative upper and lower bounds on the sizes of a typical Kakeya-type set EN and itsprincipal dilate E∗N(A0) (see Proposition 14.2).In Chapter 15 we prove the quantitative upper bound previously prescribed.Chapters 16–19 combine to establish the corresponding lower bound. Chapter 19details the actual estimation, utilizing all the smaller pieces developed in Chap-ters 16–18. These three chapters revolve around a central theme of ideas, notablythe structure imposed on the position and slope trees of a collection of two, three,or four δ-tubes, certain pairs of which are required to intersect at a given location inspace.91.3 Early history of Kakeya setsA Kakeya set (also called a Besicovitch set) in Rd is a set that contains a unit linesegment in every direction. The study of such sets spans nearly one hundred years,originating independently from the works of Abram Besicovitch and Soichi Kakeya.Besicovitch was originally interested in the question of whether or not for any givenRiemann integrable function in R2, an orthogonal coordinate system always existssuch that the two-dimensional integral is equal to the iterated integral, which mustthus be well-defined. Kakeya meanwhile was concerned with determining the smallestarea of a planar region in which a unit line segment (a “needle”) could be continuouslyrotated through 180◦, the so-called Kakeya needle problem. Surprisingly, the answersto these questions were not so disparate and provided the genesis of a long line ofinquiry that continues to this day.Besicovitch realized that his question could be answered (in the negative) if it waspossible to construct a zero Lebesgue measure planar set containing a line segmentin every direction [33]. Indeed, suppose such a set E exists and fix a coordinatesystem in R2. Define a function f so that f(x, y) = 1 if (x, y) ∈ E and if at leastone of x or y is rational, and f(x, y) = 0 otherwise. The function f is clearlyRiemann-integrable in two dimensions since its points of discontinuity comprise aset of two-dimensional Lebesgue measure zero. However, after possibly translatingE so that the x- and y-coordinates of the line segments in E parallel to the fixedcoordinate axes are irrational, we see that f is not Riemann-integrable as a functionof one variable regardless of the direction along which we choose to integrate. Infact, by the structure of E, we are guaranteed that for any direction in the plane,there is a cross-section of the function f that behaves like the characteristic functionof the rationals along the real line.The main body of Besicovitch’s work in [5] was to actually construct the requiredset E. The idea is summarized in Figure 1.4 below; in words, the idea is as follows.Begin with an equilateral triangle ABC and notice that it contains line segments ofall slopes between those of AB and AC. Now, fix some large integer n and subdividethe base BC into 2n equispaced pieces. Cut ABC into 2n thin, tall triangles with10one vertex at A and the others at subsequent points resulting from the previoussubdivision of BC. Slide these triangles along their bases to create a new figure withsmaller area, yet that still contains line segments with all the slopes of the originalABC triangle.Figure 1.4: Diagram of the first iteration of a Besicovitch-style construction of azero measure Kakeya set in the plane.Besicovitch optimized the number of cuts and the resultant reconfiguration pro-cess to show that for any ε > 0, this procedure can produce a set with area lessthan ε, while retaining all the requisite slopes from the original ABC triangle. It-erating the construction and taking the limit, Besicovitch produced a set with zero(planar) Lebesgue measure. Applying this construction individually to the union ofsix appropriately rotated copies of the original equilateral triangle results in a zero11Lebesgue measure set containing a line segment in every direction.Once such a construction is complete in the plane, it is easy to produce a zeroLebesgue measure set containing a line segment in every direction in Rd, for anyd ≥ 3. Indeed, if E denotes the zero measure planar set constructed previously, thensimply considering the set E × [0, 1]d−2 yields a valid example. Observe that sucha set is compact, and in fact always containable in a ball centred at the origin ofradius, say, 2.Besicovitch’s original construction has been simplified over the years, and thetypical construction is now quite streamlined (see Stein [42] for example). Severalalternate constructions have also been developed, most notably one by Kahane [23]which utilizes Cantor sets in the plane, and another by Besicovitch himself [7].Besicovitch did not connect the construction of his set to the Kakeya needleproblem until nearly a decade after his initial paper on the subject appeared. In fact,he was unaware of the problem due to the 1917 civil war taking place around him inRussia at the time, which effectively silenced all communication between scientistsinside the country and the rest of the scientific community [33]. In this meantime,Kakeya [24] and Fujiwara and Kakeya [18] worked on the needle problem in Japanwithout knowledge of Besicovitch’s work. They conjectured that the smallest convexplanar set within which a unit line segment could be rotated continuously through180◦ was the equilateral triangle of height 1; they also realized that one could dobetter by removing the convexity assumption. Their conjecture was soon verified byPa´l [38].In fact, it was Pa´l who pointed Besicovitch to Kakeya’s needle problem and whoalso noted how Besicovitch’s original construction could be modified to solve it [33].This lead to Besicovitch publishing the surprising solution in 1928 [6]:- For any ε > 0, there is a planar region of area less than ε within which a unitline segment can be rotated through a full 180◦.To construct such a set given an ε > 0, simply iterate Besicovitch’s originalconstruction enough times so that the resulting set E has area less than ε/2. Notice12that any tall, thin translated triangle at any iteration of the construction has at leastone side parallel to another “partner” triangle from the same stage of the construction(refer to Figure 1.4). Denote these two triangles by A1B1C1 and A2B2C2, as inFigure 1.5. Since A1B1 and A2B2 are parallel, we can take two radii D1A1 andD2A2 which form an arbitrarily small angle. Where these radii intersect, we take asector OR1R2 of radius 1. Then we can continuously move a unit line segment frominside the triangle A1B1C1, along the line D1O, through the sector OR1R2, alongthe line D2O, and through the triangle A2B2C2. Choosing a small enough angle forthe two radii D1O and D2O, we can add enough sectors to the original Besicovitchconstruction to ensure continuous movement of a needle throughout the full rangeof slopes, while retaining a total area of no more than ε.B1A1C1D1B2A2C2D2R1R2OFigure 1.5: The needle sliding component of Besicovitch’s solution to the Kakeyaneedle problem [6], as noted by Pa´l.After the publication of Besicovitch’s 1928 paper, the study of Kakeya sets fo-cused on their geometry and on the construction of similar pathological objects; forexample, sets of measure zero containing translates of all circles were constructedby Besicovitch and Rado [8], and independently by Kinney [29]. The subject wasdeveloped over the next forty-odd years by these and other mathematicians, notably13Perron, Ward, and Marstrand.Then in 1971, two seminal results appeared almost concurrently. One, due toCunningham [13], provided the most economic Kakeya-type construction and refinedBesicovitch’s answer to the original Kakeya needle problem. He showed that it waspossible to rotate a unit line segment 180◦ inside a simply connected subset of theunit circle of arbitrarily small measure. This provided a remarkable strengthening ofBesicovitch’s solution to the Kakeya needle problem, as Cunningham’s constructiondid not require the use of the needle sliding components depicted in Figure 1.5, whichresult in sets of increasingly large diameter. The other result, due to Davies [14],showed that any Kakeya set in the plane must have full Hausdorff dimension, anatural conjecture that had existed in the research community for some time. Inhigher dimensions, the analogous conjecture remains open to this day. This well-known conjecture is still most often dubbed the Kakeya conjecture.Davies’ proof of the Kakeya conjecture in the plane relied strictly on geomet-ric considerations. Soon though, it was realized that this conjecture and relatedproblems could be attacked via more analytical means, most notably by provingcertain Lp estimates on suitably defined maximal operators. Inspired largely by thework of Fefferman, notably [17], Co´rdoba reproved Davies’ result using just such amethod [11], deriving an L2(R2) estimate on an appropriate maximal operator. Hisresult hinged critically on the following geometric property (see [11] page 7, the proofof Proposition 1.2).Fact 1.4 (Co´rdoba). Let T δe (a) denote the tube of length 1 and cross-sectional radiusδ, centred at a and oriented parallel to e. For any pair of directions ek, el ∈ Sd−1,and any pair of points a, b ∈ Rd, we have the estimatesdiam(T δek(a) ∩ Tδel(b)) .δ|ek − el|,and|T δek(a) ∩ Tδel(b)| .δn|ek − el|. (1.5)This property has seen repeated application in the subsequent literature (see [9],14[46], [4], [3], for example), and has since become known colloquially as the Co´rdobaestimate.The maximal operator Co´rdoba considered was equivalent to the two-dimensionalversion of the following operator:Mδf(x) = supx∈PP∈Pδ1|P |∫P|f(y)|dy, (1.6)where f is locally integrable, and Pδ is the collection of all tubes in Rd with length1 and cross-sectional radius δ. This function has gone by many different names inthe literature over the years, including both the Kakeya maximal function and theNikodym maximal function. In the interest of compromise and historical accuracy,this function will be referred to as the restricted Kakeya-Nikodym maximal functionin the remainder. It is instructive to compare this function with the more generalform of the Kakeya-Nikodym maximal function introduced in (1.3).The study of these kinds of maximal functions, where one considers the averageof an arbitrary, locally-integrable function f over a certain set of geometric objects,dates back at least to the now classical Lebesgue differentiation theorem:Theorem 1.5. (Lebesgue) If f : Rd → R is integrable, then for almost every x wehavelimr→01|B(x, r)|∫B(x,r)f(y)dy = f(x), (1.7)where B(x, r) denotes the ball of radius r centred at x ∈ Rd.This result is of paramount significance, since it is both analogous to and ageneralization of the Fundamental Theorem of Calculus. Closely related to this, theidea of studying a maximal average of locally integrable functions over a certainset of geometric objects dates back at least to the canonical work of Hardy andLittlewood [21], near the time of Besicovitch and Kakeya’s original works previouslydiscussed. Their eponymous operator, the Hardy-Littlewood maximal function, isdefined asMHLf(x) = supr>01|B(x, r)|∫B(x,r)|f(y)|dy. (1.8)15Hardy and Littlewood’s inaugural result established that this operator was boundedon Lp(R) for all 1 < p ≤ ∞, and was also of weak-type (1,1) on R. This resultwas extended to any dimension by Wiener in a seminal 1939 work [45]; i.e. Wienerproved, among other things, that ||MHLf ||p ≤ Cp,d||f ||p for all 1 < p ≤ ∞, andthat |{x ∈ Rd : |MHLf(x)| > λ}| ≤ Cdλ−1||f ||1. These estimates can be usedto deduce the Lebesgue differentation theorem with little additional work. Theargument justifying this transition is quite robust; with minor modifications, Lpestimates on a suitably defined maximal operator can lead to differentiation theoremsover a prescribed set of geometric objects.Indeed, the balls in (1.7) and (1.8) can be replaced by cubes, by parallelopipedswith sides parallel to the coordinate axes, or any other suitably symmetric class ofobjects without changing the Lp-boundedness properties of the analogous maximalfunctions. The existence of zero measure Kakeya sets however implies that no suchbounds can hold for the Kakeya-Nikodym maximal function defined in (1.3) whenΩ is a set of directions with nonempty interior, for p < ∞. Moreover, the existenceof zero measure Kakeya sets also implies that no differentiation theorem analogousto Theorem 1.5 can hold over the collection of cylindrical tubes in Rd with arbitraryorientations and cross-sectional radius smaller than their given length L.To see this, let 0 < ε < L, and let Eε be an ε-neighbourhood of a zero measureKakeya set E ⊂ Rd constructed as the limit of sets in Besicovitch’s procedure. Thistype of Kakeya set exhibits a property called stickiness, and we say that the set Eis sticky. We will return to this concept in detail later, but for now we attempt togive just an impression of the idea.Observe that if l is any unit line segment in E, say oriented along the directionω, then if we extend l along its direction ω by at least 4 units, then this extendedsegment ˜l will fall outside the ball of radius 2 centred at the origin (recall that Ecould be constructed to be entirely contained within such a ball). Moreover, if l andl′ are two different unit line segments in E with different orientations ω and ω′, thentheir analogous extensions ˜l and ˜l′ along these directions must be disjoint. Denoteby ˜E the extended Kakeya set created by applying this extension procedure to everyunit line segment l ⊂ E, and let ˜Eε denote its ε-neighbourhood.16Now, for any x ∈ ˜Eε, note thatMΩ1Eε(x) ≥18εd−1∫ 4−4∫(−ε,ε)d−11Eε(x+ tω + sω⊥)dsdt &18,where the implicit constant is allowed to depend on L. Therefore,||MΩ||p→p &||1˜Eε||p8||1Eε ||p&∣∣∣˜Eε∣∣∣|Eε|1p, (1.9)and this holds for any 0 < ε < L. Now, Besicovitch’s construction yields |Eε| < ε,while at the same time∣∣∣˜Eε∣∣∣≥ c0 for some c0 > 0 independent of ε. Thus, if p <∞and Ω = Sd−1, the Kakeya-Nikodym operator MΩ is unbounded on Lp(Rd) by (1.9).A vast amount of work has been poured into finding δ-dependent Lp bounds forthe restricted Kakeya-Nikodym maximal operator. Such estimates have providedmyriad applications to all sorts of analytical problems; most notably, these boundscan have implications for the Kakeya conjecture, see [9], [46], [28], [27] for example.We will not explore this deep and exciting work here, but our work does focus onanother set of problems that were born out of the study of Kakeya sets and themaximal operators in (1.3) and (1.6).1.4 Background: maximal averages over lines withprescribed directionsConsider the restricted Kakeya-Nikodym maximal function, given by (1.6). What ifwe were to let δ → 0? Clearly, the limit would not make immediate sense, but noticewhat this operation tries to accomplish. Instead of averaging a function over a thintube with length 1 and cross-sectional radius δ, we would be wanting to consideraverages simply over line segments of length 1. Our object of study is then a “lowerdimensional” maximal function, the so-called directional maximal operator, definedin (1.2).17The directional maximal function is a natural generalization of the classic Hardy-Littlewood function. In fact, when d = 1, the directional maximal function reducesto (1.8). The boundedness of DΩ for finite Ω, in any dimension, follows directlyfrom the boundedness of the Hardy-Littlewood maximal operator on Lp, 1 < p ≤ ∞.Considerable work has been done on the size of the corresponding norms when Ωis a finite but otherwise arbitrary set, especially when d = 2, p = 2, most notablyin [11], [44], [15], [25], [26]. In the mid and late 1990’s, Katz showed [26] that DΩis bounded on L2(R2) with the bound .√1 + logN , where N is the cardinality ofΩ, while at the same time the lower bound &√log logN holds [25], where N = 3nand Ω is the nth iterate of the Cantor set on [0, 1]. These results establish that Lp-bounds independent of the cardinality of Ω cannot be expected to hold in general.Furthermore, these bounds together imply the unboundedness of DΩ as an operatoron L2(R2) when Ω contains a Cantor set by the same argument employed immediatelypreceding and within (1.9). This result was later generalized by Hare [22] to includesets Ω containing any Cantor set of positive Hausdorff dimension. These resultscame as somewhat of a surprise, as it had already been conjectured that cardinalityindependent bounds on DΩ should exist for Cantor sets [44]. This belief seems tohave been motivated by the fact that such cardinality independent bounds do in factexist for certain subsets of L2-functions, notably positive radial functions with weakFourier decay [44].Additionally, two famous results establish that DΩ is bounded as an operator onany Lp(Rd), 1 < p ≤ ∞, in arbitrary dimension d, for certain sets of directions Ωwith infinite cardinality. As to be expected from our discussion in Section 1.1, thesesets are required to exhibit a certain degree of lacunarity. The original results in thisdirection came from Co´rdoba and Fefferman [12], Stromberg [43], and the classicwork of Nagel, Stein, and Wainger [37]. The latter authors considered lacunary setsof the form Ω = {(θm1j , . . . , θmdj ) : j ≥ 1}, where 0 < m1 < · · · < md are fixedconstants and {θj} is a lacunary sequence with lacunarity constant 0 < λ < 1; i.e.,0 < θj+1 ≤ λθj. For such sets Ω, they showed in their 1978 paper that DΩ is boundedon all Lp(Rd), 1 < p ≤ ∞. In addition to the usual Littlewood-Paley theory, a mainfeature of their proof is the use of an almost-orthogonality principle that allows them18to control the degree of overlap between a group of averages in Fourier space (seeLemma 1, [37]). Such almost-orthogonality results reappear as a crucial part of theanalysis in all subsequent work on the subject of boundedness of these operators.In 1988, Carbery [10] utilized similar methods, substantially generalized, to showthat directional maximal operators arising from coordinate-wise lacunary sets of theform Ω = {(rk1 , . . . , rkd) : k1, . . . , kd ∈ Z+} for some 0 < r < 1 are bounded on allLp(Rd), 1 < p ≤ ∞. This result paired with the one due to Nagel, Stein, and Waingerseemed to decide the question of Lp-boundedness of the directional maximal operatorin Rd over sets Ω adhering to a certain intuitive notion of lacunarity. Extending theseideas to their limit however requires a more formal notion of generalized lacunarity,in addition to the appropriate almost-orthogonality results.In the plane, this notion was sufficiently generalized by Sjo¨gren and Sjo¨lin [41] in1981, establishing that so-called lacunary sets of finite order gave rise to boundeddirectional maximal operators on Lp(R2)-space, 1 < p ≤ ∞. On the real line, theydefined a lacunary set of order 0 as a singleton, and a lacunary set of order N ≥ 1 asa successor of a lacunary set of order N − 1, where the successor of a set is definedas follows. For any closed sets A,A′ ⊂ R of Lebesgue measure zero, A′ is a successorof A if there exists a constant c > 0 such that x, y ∈ A′, with x 6= y, implies that|x − y| ≥ c · distA(x). Here, distA(x) denotes the usual distance between two sets.Using such a definition, sets of the form{N∑k=1λikk : ik ∈ Z, 1 ≤ k ≤ N}∪ {0} (1.10)are lacunary of order N . Once this notion is defined on the real line, it may belifted and applied directly to sets of directions in the plane, Ω ⊆ S1. In this way,Sjo¨gren and Sjo¨lin were able to generalize the results of [37], as well as to prove thatL2-boundedness cannot be expected to hold for direction sets Ω ⊂ S1 correspondingto a classical Cantor set.In 2003, Alfonseca, Soria, and Vargas published an L2-almost-orthogonality resultfor directional maximal operators in the plane [2], shortly thereafter generalized by19Alfonseca to any Lp-space [1]. Essentially, what was recognized was that if Ω0 ⊂Ω ⊆ [0, pi4 ) is an ordered subset of angles,pi4= θ0 > θ1 > θ2 > . . . ,then the Lp-norm of the directional maximal operator over Ω is controlled by theLp-norm of DΩ0 and the supremum of the Lp-norms of DΩj , where for each j ≥ 1,one defines Ωj = Ω ∩ [θj, θj−1). These results substantially generalized the originalalmost-orthogonality principle proved by Nagel, Stein, and Wainger and played acritical role in the work of Bateman [3].These almost-orthogonality results and the study of lacunarity were not the onlyfoci of research on directional maximal operators. As previously mentioned, concur-rent work had been taking place that studied the behaviour of the Lp-norms of DΩwhen Ω is given by a Cantor subset of directions [15], [44], [25], [26]. This researchaimed to uncover the middle ground between lacunary directional maximal opera-tors, and those arising from a set of directions Ω with nonempty interior. Indeed, dueto the existence of zero measure Kakeya sets, and by the same argument that appearsin and preceding inequality (1.9), it is an immediate consequence that all operatorsDΩ arising from Ω with nonempty interior are unbounded on Lp(Rd), p 6= ∞.In the plane, Bateman and Katz finally settled the issue of Lp-boundedness ofdirectional maximal operators arising from a Cantor set of directions Ω for a genericp ∈ [1,∞) in their 2008 publication [4]. They showed that not only is such anoperator DΩ unbounded on Lp(R2), p 6= ∞, but more strongly that such a set ofdirections Ω admits Kakeya-type sets. Precisely, they showed that for any n ≥ 1,there is a union of n parallelograms in R2 of dimensions 1 × 1n , and with slopes ofthe longest sides contained in the standard middle-thirds Cantor set, so that∣∣∣∣∣n⋃j=1Pj∣∣∣∣∣.1log n, while∣∣∣∣∣n⋃j=12Pj∣∣∣∣∣&log log nlog n. (1.11)Notice that a set that obeys (1.11) is indeed of Kakeya-type according to our Defi-nition 1.1.20Bateman and Katz’s construction is probabilistic. In particular, they show thatthe first inequality in (1.11) holds for a typical collection of suitable parallelogramswith respect to a certain uniformly distributed probability measure. This result relieson an ingenious application of a seminal theorem from the theory of percolation ontree structures due to Russ Lyons [34], [35]. Their work represents the first time suchan idea has been applied to the study of Kakeya-type sets and related objects. Thenew results that appear in this document will rely critically on the same idea (seeChapters 11 and 15).Shortly after [4] appeared, Bateman published a generalization of their result [3]that provided a complete characterization of the Lp-boundedness of directional max-imal operators in the plane for an arbitrary set of directions Ω ⊆ S1. The proofutilized the same techinques as in [4], with the addition of some necessary complex-ities to account for the totally general structure of the direction set Ω. Introducedoriginally perhaps as a convenient way to formally apply Lyons’ percolation result,the inherent “tree structure” naturally associated to an arbitrary set of directionsplayed a more vital role in Bateman’s work. This type of encoding allows for arich and flexible structure capable of accommodating both the generic nature of theproblem and the specific combinatorics and analysis required to establish (1.11) inthe general context.1.5 Kakeya-type sets and the property of sticki-nessIf we consider once again Besicovitch’s construction of a zero measure Kakeya set,described near the beginning of Section 1.3 and partially depicted in Figure 1.4, aninteresting observation can be made; indeed, we have already appealed to this obser-vation when proving the unboundedness of the Kakeya-Nikodym maximal operatorover Sd. Although the many thin triangles in that construction are translated as tooverlap near a common core, by virtue of the disparate directions they each contain,these triangles are disjoint sufficiently far away from that core. Put more generically,21if many thin tubes with distinct directions intersect at a common locus, then thesetubes will become mutually disjoint outside a large enough ball surrounding this lo-cus. The necessary size of such a ball is dependent on both the cross-sectional widthof these tubes, and the amount of separation between their directions. A collectionof tubes that exhibits this property are said to be sticky, referring to the idea thatthey should all “stick together” near a common core. Such a collection is picturedin Figure 1.6.small measurelarge measureFigure 1.6: A sticky collection of tubes⋃Pt, along with their extensions⋃A0Pt inR3.Wolff was the first to formally recognize that zero measure Kakeya sets mustenjoy this property; see Property (∗) in [46]. More specifically, if Kakeya sets are tohave small measure, then a δ-thickening of the set must contain many pieces thatlook like the “small measure,” right hand side of Figure 1.6; i.e., contain many tubesthat overlap for much of their length. This can only be so if both the centres of thesetubes and their orientations are nearly the same. In this case, as long as the tubesare given distinct orientations, an extension of these tubes by an appropriate amountmust result in a set of relatively large measure, as in the left hand side of Figure 1.6.Indeed, if we again consider the Besicovitch construction in Figure 1.4, althoughthe original triangle can be chosen to have, say, unit area, the resulting set aftern iterations of the construction can be made to have area less than about 1/ log n.However, extending each subdivided triangle along their respective diameters willresult in a set with approximately unit area, independent of n.The reader will recognize that the phenomenon we have been describing has22already been laid down in quantitative terms in Definition 1.1, and exploited ana-lytically to show why the existence of Kakeya-type sets implies unboundedness ofvarious associated maximal operators, as in the argument leading to (1.9). Indeeed,the condition (1.1) is a generic formulation of the idea of stickiness for a family oftubes with orientations arising from a given set of directions Ω. We see then thatthe idea of a Kakeya-type set is motivated by the property of stickiness exhibitedby traditional Kakeya sets of zero measure; Kakeya-type sets in fact preserve thisproperty as their defining feature.The idea of stickiness is not restricted to collections of tubes. Although similarnotions can be defined for essentially any collection of common, thickened lower di-mensional objects in Euclidean space, sticky collections of curves, circles, and sphereshave been of particular interest [8], [29], [40], [30]. We will discuss further the ideaof stickiness and its interplay with work developed in this document in Chapter 20.23Chapter 2Finite order lacunarityAs noted in Section 1.4, the concept of finite order lacunarity plays a fundamentalrole in the study of planar Kakeya-type sets and associated directional maximaloperators. The results of Nagel, Stein, Wainger [37], and Carbery [10] suggest that itcontinues to play a similar role in higher dimensional space. The existing literatureon the subject embodies several different notions of Euclidean lacunarity both insingle and general dimensions, see in particular [3, 10, 39, 41]. The present chapteris devoted to a discussion of the definitions to be used in the remainder of the paper.The concepts introduced here will be revisited in Section 3.7, using the language oftrees. The interplay of these two perspectives is essential to the proof of Theorem1.3.2.1 Lacunarity on the real lineDefinition 2.1 (Lacunary sequence). Let A = {a1, a2, . . .} be an infinite sequence ofpoints contained in a compact subset of R. Given a constant 0 < λ < 1, we say thatA is a lacunary sequence converging to α with constant of lacunarity at most λ, if|aj+1 − α| ≤ λ|aj − α| for all j ≥ 1.Definition 2.2 (Lacunary sets). In R, a lacunary set of order 0 is a set of cardinality24at most 1; i.e., either empty or a singleton. Recursively, given a constant 0 < λ < 1and an integer N ≥ 1, we say that a relatively compact subset U of R is a lacunaryset of order at most N with lacunarity constant at most λ, and write U ∈ Λ(N, λ),if there exists a lacunary sequence A with lacunarity constant ≤ λ with the followingproperties:- U ∩ [sup(A),∞) = ∅, U ∩ (−∞, inf(A)] = ∅,- For any two elements a, b ∈ A, a < b such that (a, b) ∩ A = ∅, the set U ∩ [a, b) ∈Λ(N − 1, λ).The order of lacunarity of U is exactly N if U ∈ Λ(N, λ) \ Λ(N − 1, λ). A lacunarysequence A obeying the conditions above will be called a special sequence and its limitwill be termed a special point for U .For any fixed N and λ, the class Λ(N, λ) is closed under containment, scalaraddition and multiplication; these properties, summarized in the following lemma,are easy to verify.Lemma 2.3. Let U ∈ Λ(N, λ). Then(i) V ∈ Λ(N, λ) for any V ⊆ U .(ii) c1U + c2 ∈ Λ(N, λ) for any c1 6= 0, c2 ∈ R.Proof. For (i), U ∈ Λ(N, λ) means that there exists a lacunary sequence A withlacunarity constant ≤ λ such that U ∩ [sup(A),∞) = ∅, U ∩ (−∞, inf(A)] = ∅, andU ∩ [a, b) ∈ Λ(N − 1, λ) for any a, b ∈ A with a < b and (a, b)∩A = ∅. Since V ⊆ U ,the same set of conditions is satisfied exactly with U replaced by V .To prove (ii), we note that c1A + c2 can serve as special sequence for c1U + c2,if A is a special sequence for U . Since U ∩ [a, b) ∈ Λ(N − 1, λ) if and only if(c1U + c2)∩ [c1a+ c2, c1b+ c2) ∈ Λ(N − 1, λ) with a, b ∈ A, a < b, (a, b)∩A = ∅, thelemma follows.The sets of interest to us are those that are generated by finite unions of sets ofthe form described in Definition 2.2.25Definition 2.4 (Admissible lacunarity of finite order and sublacunarity). We saythat a relatively compact set U ⊆ R is an admissible lacunary set of finite order ifthere exist a constant 0 < λ < 1 and integers 1 ≤ N1, N2 < ∞ such that U can becovered by N1 lacunary sets of order ≤ N2, each with lacunarity constant ≤ λ. If Udoes not satisfy this criterion, we call it sublacunary.2.1.1 Examples(a) A standard example of a lacunary set of order 1 and lacunarity constant λ ∈ (0, 1)is U = {λj : j ≥ 1}, or any nontrivial subsequence thereof. Indeed U is itself alacunary sequence, and hence its own special sequence.A general lacunary set of order 1 need not always be a lacunary sequence. For ex-ample {2−2j±4−2j : j ≥ 1} is lacunary of order 1 relative to the special sequence{2−j : j ≥ 1}. Despite this, lacunary sequences are in a sense representative ofthe class Λ(1, λ), since any set in Λ(1, λ) can be written as the union of at mostfour lacunary sequences with lacunarity constant ≤ λ. By Lemma 2.3, the set{aλj + b : j ≥ 1} is lacunary of order at most 1 for any unit vector (a, b).(b) In general, given an integer k ≥ 1 and constants M1 ≤ M2 ≤ · · · ≤ Mk withM1 ≥ max(2, k − 1), the setU ={M−j11 +M−j22 + · · · +M−jkk : 0 ≤ j1 ≤ j2 ≤ · · · ≤ jk}is lacunary of order k and has lacunarity constant ≤M−11 . The special sequencecan be chosen to be A = {M−j1 : j ≥ 1}.(c) A set that is dense in some nontrivial interval, however small, is sublacunary.For example, dyadic rationals of the form { k2m : 0 ≤ k < 2m} for a fixed m canbe written as a finite union of lacunary sequences with a given lacunarity λ, butthe number of sequences in the union grows without bound as m → ∞. ByLemma 2.3, a set that contains an affine copy of { k2m : 0 ≤ k < 2m} for every mis sublacunary.26(d) The set U = {2−j + 3−k : j, k ≥ 0} can be covered by a finite union of setsin Λ(2, 12). For instance the two subsets of U where k ln 3 ≤ (j − 1) ln 2 andk ln 3 ≥ j ln 2 respectively are each lacunary of order 2, with {3−k} and {2−j}being their respective special sequences. The complement, where (j − 1) ln 2 ≤k ln 3 ≤ j ln 2, contains at most one k per j, and is a finite union of lacunary setsof order 1.(e) A slight variation of the above example: {2−j + (qj − 2−j)3−k : j, k ≥ 0}, where{qj} is an enumeration of the rationals in [ 910 , 1], leads to a very different con-clusion. This set contains {qj}, and is hence sublacunary, even though the setmay be viewed as a special sequence {2−j} with collections of lacunary sequencesconverging to every point of it. This example illustrates the relevance of the re-quirement that the lower order components of Λ(N, λ) lie in disjoint intervals ofR.(f) Given any 0 < λ < 1 and m > 0, there is a constant C = C(λ,m) such thatfor any unit vector (a, b), the set Ua,b = {aλj + bλmj : j ≥ 1} can be covered byC sets in Λ(1, λ). We prove an analogous statement along these lines in Section2.2.1, see example (a).(g) Given any 0 < λ < 1, m ∈ Q ∩ (0,∞), there is a constant C = C(λ,m) suchthat for any unit vector (a, b), the setUa,b = {ujk = aλj + bλmk : j, k ≥ 1}can be covered by at most C lacunary sets of order at most 2. This is clear for(a, b) = (1, 0) or (0, 1), with the order of lacunarity being 1. For ab 6= 0, thereare four possibilities concerning the signs of a and b. We deal with a > 0 andb < 0, the treatment of which is representative of the general case. The set Ua,bis decomposed into three parts:Va,b ={ujk ∈ U : aλj + bλmk ≥ aλj+1},Wa,b ={ujk ∈ U : aλj + bλmk < bλm(k+1)},27Za,b = Ua,b \[Va,b ∪Wa,b].Then for every fixed j, the set Va,b∩[aλj+1, aλj) is an increasing lacunary sequencewith constant ≤ λm, converging to aλj. An analogous conclusion holds forWa,b ∩ [bλmk, bλm(k+1)). Thus Va,b and Wa,b are both lacunary of order 2, withtheir special sequences being A = {aλj} and A = {bλmk} respectively. Forujk ∈ Za,b, the indices j and k obey the inequality−ab(1 − λ) < λmk−j ≤ −ab(1 − λm)−1.Since m is rational, the values of mk − j range over rationals of a fixed demon-imator (same as that of m). The inequality above therefore permits at most Csolutions of mk − j, the constant C depending on λ and m, but independentof (a, b). Thus Za,b is covered by a C-fold union of subsets, each consisting ofelements ujk = λj(a + bλmk−j) for which mk − j is held fixed at one of thesesolutions. Each such set is lacunary of order 1 with lacunarity ≤ λ.2.1.2 Non-closure of finite order lacunarity under algebraicsumsAn important aspect of the class of admissible lacunary sets of finite order is that itis not closed under set-algebraic operations, as we establish in the example furnishedbelow. This feature, perhaps initially counterintuitive, is the main inspiration forthe definition of higher dimensional lacunarity provided in the next subsection.Example: Let Nj ↗ ∞ be a fast growing sequence, and Mj = 2mj a slowergrowing one, so thatMj < Nj −Nj−1. (2.1)28For instance, Nj = 2j2 and Mj = 2j will do. For every j ≥ 1 and 1 ≤ k ≤Mj = 2mj ,set qjk = 2−Nj(1 + k2−mj), and defineUj = {2−Nj+k + qjk : 1 ≤ k ≤Mj}, U =∞⋃j=1Uj, V = {−2−j : j ≥ 1}.An element of Uj of the form 2−Nj+k+qjk lies in the dyadic interval [2−Nj+k, 2−Nj+k+1),and for a given k, is the only element of Uj in this interval. Further, Uj ⊆ [2−Nj+1, 2−Nj+Mj+1),hence by the relation (2.1), Uj ∩ Uj′ = ∅ if j 6= j′. Thus U ∈ Λ(1, 12), since for anyi ≥ 1, the set U∩[2−i, 2−i+1) is either empty or a single point. Clearly V is a lacunarysequence, hence V ∈ Λ(1, 12) as well, being its own special sequence. On the otherhand,U + V ⊇∞⋃j=1{qjk : 1 ≤ k ≤Mj}.In other words, U + V contains an affine copy of the dyadic rationals of the form{k2−mj : 1 ≤ k ≤ 2mj} in [0, 1], for every j. As discussed in example (c) in Section2.1.1, U + V is sublacunary.The counterexample above illustrates the sensitivity of lacunarity on ambientcoordinates, and precludes a higher dimensional generalization of this notion thatrelies on componentwise extension. For instance, the two-dimensional set U × V(with U , V as above) has lacunary coordinate projections in the current system ofcoordinates, but there are other directions of projection, for instance the line of unitslope, along which the projection of this set is much more dense.2.2 Finite order lacunarity in general dimensionsLet V be a d-dimensional affine subspace of an Euclidean space Rn, n ≥ d. Given abase point a of V and an orthonormal basis B = {v1, . . . ,vd} of the linear subspace29V− a, we define the projection mapspij = pij[a,B] : V→ R, via x = a +d∑i=1xivi → xi = pii(x), 1 ≤ i ≤ d. (2.2)Definition 2.5 (Admissible lacunarity and sublacunarity of Euclidean sets). Let Ube a relatively compact subset of V.(i) We say that the set U is admissible lacunary of order at most N (as an Eu-clidean subset of V) with lacunarity constant at most λ < 1 if there exists aninteger R ≥ 1 satisfying the following property: for any choice of basis B andbase point a, and each 1 ≤ j ≤ d, the projected setpij(U) = {pij(x) : x ∈ U} ⊆ Rcan be covered by R members of Λ(N, λ), with the class Λ(N, λ) as described inDefinition 2.2. The projection pij depends on a and B via (2.2). The collectionof sets U that obey these conditions for a given choice of N, λ and R will bedenoted by Λd(N, λ,R;V).(ii) The set U is called sublacunary in V if it is not admissible lacunary of finiteorder; i.e., if for any λ < 1 and integers N,R ≥ 1 there exists a choice ofbasis B and an index 1 ≤ j ≤ d such that pij(U) cannot be covered by any R-fold union of one-dimensional lacunary sets of order at most N and lacunarityconstant at most λ.Remarks:- An equivalent formulation of the definition of U ∈ Λd(N,R, λ;V) is that for anyline L in V (and indeed in Rd+1 as we will soon see in Lemma 2.6), the projectionof U onto L is coverable by at most R sets in Λ(N, λ).- The choice of base point in V is not important in this definition, since pij[a,B](U)is a translate of pij[a′,B](U) for any a, a′ ∈ V. Thus pij[a,B](U) ∈ Λ(N, λ) if andonly if pij[a′,B](U) ∈ Λ(N, λ).30- The definition is also invariant under rotation in Rn; if O is an orthogonal trans-formation of Rn, then U ∈ Λd(N, λ,R;V) if and only if O(U) ∈ Λd(N, λ,R;O(V)).- The choice of rotation B within V is however critical. It is not possible to havenecessary and sufficient implications like the ones above for two arbitrary choicesof bases B and B′. We provide examples below. Henceforth, we will refer to thechoice of a pair ϕ = (a,B) as a system of coordinates, with the main focus on B.- Loosely speaking, the order of lacunarity N may be viewed as the number ofindependent parameters required to describe Ω. The examples in the next sectionwill substantiate this statement.- If U ∈ Λd(N, λ,R;V), then the order of lacunarity of piLU cannot exceed N forgeneric lines L, where piL denotes the projection onto L. However, there may existmany lines onto which the projections are much sparser than what the order oflacunarity suggests, see for example (2.8).Before proceeding to examples, we check the definition for consistency if U is a subsetof several affine subspaces.Lemma 2.6. Let U ⊆ V be as above. Then for any choice of N,R, λ, the setU ∈ Λd(N,R, λ;V) if and only if U ∈ Λn(N,R, λ;Rn).Proof. The “if” implication is clear, so we consider the converse. Without loss ofgenerality, we may choose V = {1}×Rn−1. Given any unit vector ω = (ω1, · · · , ωn) ∈Rn with 0 < |ω1| < 1, let L denote the line through the origin in Rn pointing in thedirection of ω. Let L′ denote the projection of L on V, so that L′ = {e1+sω′ : s ∈ R},where e1 is the first canonical basis vector in Rn, and ω′ = (0, ω2, · · · , ωn). Thedesired conclusion follows from the claim thatthe sets pi(U) and pi′(U) are affine copies of each other, (2.3)where pi(U) and pi′(U) denote the scalar projections onto L and L′, measured fromthe origin and from (1, 0, · · · , 0) respectively. Indeed, Lemma 2.3 then permits us toextend known lacunarity features of the former directly to the latter.31To establish (2.3), it suffices to note that for any x ∈ Rn,pi(x) = (x · ω)ω, and pi′(x) = x · ω′|ω′|2ω′.The choice of V, ω and ω′ yield the relations (x−y) ·ω = (x−y) ·ω′ for any x, y ∈ U ,hence the above expressions imply that|pi(x) − pi(y)| = |ω′||pi′(x) − pi′(y)|,which is the desired conclusion.2.2.1 Examples of admissible lacunary and sublacunary setsin Rd(a) A set of the form considered by Nagel, Stein and Wainger [37], such asU = {γ(θj) : j ≥ 1}, where γ(t) = (tm1 , · · · , tmd) (2.4)is admissible lacunary of order 1. Here 0 < m1 < · · · < md are fixed constants,and 0 < θj+1 ≤ λθj, for some 0 < λ < 1 and all j. Critical to this verificationare the following two properties of U appearing in [37, Lemma 4]:- There is a constant C1 = C1(m1, · · · ,md) obeying the following requirement.For any unit vector ξ = (ξ1, · · · , ξd) in Rd, the set N of positive integers canbe decomposed into C1 disjoint consecutive intervals {Ns}; for every s, thereexists r(s) ∈ {1, · · · , d} such thatmax1≤r≤d|θmrj ξr| =∣∣θmr(s)j ξr(s)∣∣ for all j ∈ Ns. (2.5)The composition of Ns depends on ξ.- Further for any c > 0, there is a constant C2 = C2(c,m1, · · · ,md) independent32of ξ and Ns so thatmaxr∈{1,··· ,d}r 6=r(s)∣∣θmrj ξr∣∣ < c∣∣θmr(s)j ξr(s)∣∣ (2.6)for all but C2 integers j ∈ Ns.Assuming these two facts, the claim of lacunarity is established as follows. Usingthe definition of Ns in (2.5), the set U can be decomposed into C1 pieces Us, whereUs = {γ(θj) : j ∈ Ns}. Fix a constant R such that 2dλm1R−1 < 1. If j′ > jare two integers in Ns that are at least R-separated and for both of which (2.6)holds with c = 12d , then∣∣d∑r=1ξrθmrj′∣∣ ≤ d∣∣ξr(s)θmr(s)j′∣∣ ≤ d(λj′−j)mr(s)∣∣ξr(s)θmr(s)j∣∣≤ 2d(λR)m1∣∣d∑r=1ξrθmrj∣∣ < λ∣∣d∑r=1ξrθmrj∣∣.(2.7)Thus each Us is the union of at most R lacunary sequences of lacunarity < λ,together with the C2 points where (2.6) fails.(b) A set of the form considered by Carbery [10], i.e.,U = {Γk = (λk1 , · · · , λkd) : k = (k1, · · · , kd) ∈ Nd} (2.8)is admissible lacunary of order d. We prove this by induction on d. The initializ-ing step for d = 2 has been covered in example (g) of Section 2.1.1. For a generald and after splitting U into d! pieces, we may assume that k1 ≤ k2 ≤ · · · ≤ kd.Given any unit vector ξ = (ξ1, · · · , ξd) ∈ Rd, we writeU =d⋃s=1Us with Us = {Γk : k ∈ Nds}, whereNds ={k ∈ Nd :∣∣λksξs∣∣ = max1≤r≤d∣∣λkrξr∣∣}.33Depending on the signs of λksξs and Γk · ξ − λksξs, each Nds can be decomposedinto four parts. Their treatments are similar with trivial adjustments, so wefocus on the subset of Nds whereλksξs > 0 and∑r 6=sλkrξr ≥ 0,continuing to call this subset Nds to ease notational burden. One last splitting isneeded; for a constant A to be specified shortly, we writeNds = Nds,1 ∪ Nds,2, where Nds,1 = {k ∈ Nds : λksξs > A|λkrξr| for all r 6= s}.For k ∈ Nds,1,λksξs ≤ Γk · ξ < λksξs(1 + dA−1)< λks−1ξs, (2.9)where the last inequality follows for a suitable choice of A. We argue that{ξsλks : ks ≥ 1} may be viewed as a special sequence for {Γk · ξ : k ∈ Nds,1}.Indeed, if ks is fixed, then (2.9) shows that{Γr · ξ : r ∈ Nds,1} ∩ [ξsλks , ξsλks−1) = {Γr · ξ : r ∈ Nds,1, rs = ks}⊆ ξsλks +{∑r 6=sλkrξr : kr ∈ N, r 6= s}.By the induction hypothesis, there is a constant R independent of ξ such thatthe set on the right hand side above is coverable by at most R sets in Λ(d−1;λ).Hence {Γk : k ∈ Nds,1} is admissible lacunary of order d.We turn to the complementary set Nds,2. After decomposing Nds,2 into (d − 1)subsets, we may fix an index ` such that|λk`ξ`| ≤ λksξs ≤ A|λk`ξ`| (2.10)on Nds,2. Without loss of generality let ` ≥ s. The number of possible values ofk` − ks obeying (2.10) is at most a fixed constant C depending on A (hence λ34and d), but independent of ξ. Thus Nds,2 may be written as the C-fold union ofsubsets indexed by c, where the subset identified by c contains all k ∈ Nds,2 withthe property that k` − ks = c ≥ 0. For k in such a subset,Γk · ξ = (ξs + λcξ`)λks +∑r 6=`,sλkrξr.Since the number of summands in the linear combination above is (d − 1), theinduction hypothesis dictates that {Γk : k ∈ Nds,2} is admissible lacunary of order(d− 1), completing the proof.(c) A curve in Rd is sublacunary. So is a Cantor-like subset of it, by Theorem 1.2.(d) If U and V are the lacunary sets of order 1 constructed in Section 2.1.2, the setU × V is sublacunary. Indeed, after a rotation of angle pi4 one of the coordinateprojections turns out to be a constant multiple of U+V . We have seen in Section2.1.2 that this last set is sublacunary on R.2.3 Finite order lacunarity for direction setsGiven two sets Ω1,Ω2 ⊆ Rd+1 \ {0}, we say that Ω1 ∼ Ω2 if{ω|ω|: ω ∈ Ω1}={ω|ω|: ω ∈ Ω2}.The binary relation ∼ is clearly an equivalence relation among sets in Rd+1 \ {0}.An equivalence class of ∼ is, by definition, a direction set. By a slight abuse ofnomenclature, we will refer to a set Ω ⊆ Rd+1 \ {0} as a direction set to mean theequivalence class of ∼ that contains Ω. Clearly the maximal operators DΩ and MΩ, aswell as the admittance of Kakeya-type sets (as in Definition 1.1), remain unchangedfor all members of this equivalence class.Certain modifications are necessary to extend the notion of lacunarity from Eu-clidean sets to direction sets, in view of the latter’s scale invariance. Given a direction35set Ω ⊆ Rd+1 \ {0}, we denote by CΩ the cone generated by this set of directions,namelyCΩ := {rω : r > 0, ω ∈ Ω}. (2.11)Definition 2.7. Let Ω ⊆ Rd+1 \ {0} be a direction set, with CΩ as in (2.11).(i) Given an integer N and a positive constant λ < 1, we say that Ω is admissiblelacunary as a direction set with order at most N and lacunarity at most λ ifthere exists an integer R such that U ∈ Λd(N, λ,R;V) in the sense of Definition2.5, for every hyperplane V at unit distance from the origin and every relativelycompact subset U of CΩ ∩ V. The collection of direction sets that obey theseconditions for a given N, λ and R will be denoted by ∆d(N, λ,R).(ii) A direction set Ω ⊆ Ω0 failing this property is termed a sublacunary directionset. Thus Ω is sublacunary as a direction set if for any choice of integersN,R and positive constant λ < 1 there is a tangential hyperplane V of the unitsphere, a relatively compact subset U of CΩ ∩ V and a line L in V such thatthe projection of U along L cannot be covered by any R-fold union of sets inΛ(N, λ).2.3.1 Examples of admissible lacunary and sublacunary di-rection sets(a) A direction set Ω of the form considered by Nagel, Stein and Wainger [37],Ω = {uj = (γ(θj), 1) : j ≥ J}is admissible lacunary of order 1. Here the function γ and the sequence θj areas described in example (a) of Section 2.2.1. Thus Ω is parameterized by thepositive constants m1 < m2 < · · · < md. We set md+1 = 0. To verify the claim,we choose V = {x ∈ Rd+1 : x · η = 1} for some unit vector η, so thatCΩ ∩ V ={vj =ujuj · η: uj ∈ Ω}.36Fix a unit vector ω = (ω′, ωd+1) ∈ Rd+1, and let piω denote the scalar projectiononto ω; i.e., piω(v) = v · ω. As required by Definitions 2.7 and 2.5 and in view ofLemma 2.6, we aim to show that that there is a large constant R (independentof V) for which any relatively compact subset of piω(CΩ∩V) can be covered by Rmembers of Λ(1;λ). By the property (2.5) of Ω, we first decompose the integersinto a bounded number C1 of disjoint intervals (C1 independent of ω and η), oneach of which there exists an index 1 ≤ r ≤ d+ 1 such thatmax1≤i≤d+1|θmij ηi| = |θmrj ηr|. (2.12)Let us denote by Nr[η] one of the subintervals for which (2.12) holds. For j ∈Nr[η],piω(vj) −ωrηr= ξ · ujηr (η · uj), where ξ = (ξ1, · · · , ξd+1) ∈ Rd+1 (2.13)with ξk = ωkηr − ωrηk, so that ξr = 0. Our goal is to show that for j ∈ Nr[η],the sequence on the right hand side above can be covered by an R-fold union oflacunary sequences converging to 0.Using (2.5) again, we decompose Nr[η] into at most C1 pieces, of the formNrs[η, ξ] = Nr[η] ∩ Ns[ξ]. Since ξr = 0, we conclude that Nr[ξ] = ∅; hences 6= r. By property (2.6), for every c > 0, there are at most a bounded numberC2 = C2(c) indices j ∈ Nrs[η, ξ] for which at least one of the inequalitiesmaxi6=r|θmij ηi| < c|θmrj ηr|, maxi6=s |θmij ξi| < c|θmsj ξs| (2.14)fails.First suppose s > r. Choosing two integers j, j′ ∈ Nrs[η, ξ] with j′ − j ≥ R forboth of which the constraints in (2.14) hold, we follow the steps laid out in (2.7),37obtaining from (2.13)[∣∣∣∣piω(vj′) −ωrηr∣∣∣∣] [∣∣∣∣piω(vj) −ωrηr∣∣∣∣]−1= ξ · uj′ξ · uj· η · ujη · uj′≤[d|ξs|θmsj′12 |ξs|θmsj]·[d|ηr|θmrj12 |ηr|θmrj′]≤ 4d2(θj′θj)ms−mr≤ 4d2λR(ms−mr).If R is selected large enough to satisfy 4d2λR(ms−mr) < λ, then for j ∈ Nrs[η, ξ]the sequence on the right hand side of (2.13) can be covered by the union ofR lacunary sequences converging to zero, excluding the C2 points where (2.14)fails. For s < r, the same calculation above can be replicated for j′ < j withj′ − j < −R. Thus in this case the sequence in (2.13) grows as j increases, andhence has to be finite by the assumption of relative compactness. Nonetheless,this finite sequence is still coverable by a lacunary sequence going to zero, thistime in reverse order of j. In either event, we have decomposed the set {piω(vj) :j ∈ Nrs[η, ξ]} into R lacunary sequences of lacunarity λ, proving the claim.(b) A direction set of the type studied in [10], namelyΩ = {(Γk, 1) : 0 ≤ k1 ≤ k2 ≤ · · · ≤ kd},(with Γk as in (2.8)) is admissible lacunary of order d. This is proved along linessimilar to the example above, using methods already explained in examples (g)and (b) of Section 2.1.1 and 2.2.1 respectively; we omit the details here.(c) A curve in Rd+1 is sublacunary as a direction set.(d) For sets U , V as constructed in Section 2.1.2, the direction set Ω = {1}×U ×Vis sublacunary, since U × V is sublacunary as an Euclidean set (see example (d)in Section 2.2.1).(e) Let {q` : ` ≥ 1} be an enumeration of the rationals on any nontrivial interval,38say on [12 ,23 ]. A direction set of the type considered by Parcet and Rogers [39,Example 1 on page 4], such asΩ = {(q`2−`, 2−`, 1) : ` ≥ 1}is sublacunary, even though the one-dimensional coordinate projections in thecurrent coordinate system are lacunary of order at most 1. Choosing V = {x2 =1}, we find thatCΩ ∩ V = {(q`, 1, 2`) : ` ≥ 1}.The order of lacunarity of the x1-projection grows without bound as we chooseincreasingly large compact subsets of CΩ ∩ V.(f) We also mention another example considered by Parcet and Rogers [39, Example2 on page 4]. Given the canonical orthonormal basis {e1, e2, e3} of R3, let us fixanother orthonormal basis {e1, e′2, e′3} with span{e2, e3} = span{e′2, e′3} and e′3lying in the first quadrant determined by e2 and e3. The direction set underconsideration is Ω = {u` : ` ≥ 1}, where u` is a sequence of vectors satisfyingu` · e′2 = q`u` · e1 for some enumeration of rationals {q`} in an interval. Thelast condition does not completely specify u`, hence the direction set so definedis not unique (further restrictions are imposed in [39]), but regardless of anysubsequent choice Ω is sublacunary. Choosing V = {x1 = 1}, we observe thatCΩ ∩ V ={u`u` · e1: ` ≥ 1}.Projecting CΩ∩V in the direction e′2, we find that the projected set is {q` : ` ≥ 1},which is not lacunary of finite order.39Chapter 3Rooted, labelled treesAs in Bateman and Katz’s work [4], [3], the language of rooted, labelled trees remainsthe vehicle of choice for construction of Kakeya-type sets. We explore the basicterminology of trees and state the relevant facts in Sections 3.1 and 3.6 below. Thisconstitutes only a very small part of the literature on such objects, but will sufficefor our purposes. These objects are treated comprehensively in the text by Lyonsand Peres [36].3.1 The terminology of treesAn undirected graph G := (V , E) is a pair, where V is a set of vertices and E is asymmetric, nonreflexive subset of V ×V , called the edge set. By symmetric, here wemean that the pair (u, v) ∈ E is unordered; i.e. the pair (u, v) is identical to the pair(v, u). By nonreflexive, we mean E does not contain the pair (v, v) for any v ∈ V .A path in a graph is a sequence of vertices such that each successive pair ofvertices is a distinct edge in the graph. A finite path (with at least one edge) whosefirst and last vertices are the same is called a cycle. A graph is connected if for eachpair of vertices v 6= u, there is a path in G containing v and u. We define a tree tobe a connected undirected graph with no cycles.All our trees will be of a specific structure. A rooted, labelled tree T is one whose40vertex set is a nonempty collection of finite sequences of nonnegative integers suchthat if 〈i1, . . . , in〉 ∈ T , then(i.) for any k, 0 ≤ k ≤ n, 〈i1, . . . , ik〉 ∈ T , where k = 0 corresponds to the emptysequence, and(ii.) for every j ∈ {0, 1, . . . , in}, we have 〈i1, . . . , in−1, j〉 ∈ T .We say that 〈i1, . . . , in−1〉 is the parent of 〈i1, . . . , in−1, j〉 and that 〈i1, . . . , in−1, j〉 isthe (j + 1)th child of 〈i1, . . . , in−1〉. If u and v are two sequences in T such that uis a child of v, or a child’s child of v, or a child’s child’s child of v, etc., then we saythat u is a descendant of v (or that v is an ancestor of u), and we write u ⊂ v (seethe remark below). If u = 〈i1, . . . , im〉 ∈ T , v = 〈j1, . . . , jn〉 ∈ T , m ≤ n, then theyoungest common ancestor of u and v is the vertex in T defined byD(u, v) = D(v, u) :=∅, if i1 6= j1〈i1, . . . , ik〉 if k = max{l : il = jl}.(3.1)One can similarly define the youngest common ancestor for any finite collection ofvertices.Remark: At first glance, using the notation u ⊂ v to denote when u is a descendantof v may seem counterintuitive, since u is a descendant of v precisely when v is asubsequence of u. However, we will soon be identifying vertices of rooted labelledtrees with certain nested families of cubes in Rd. Consequently, as will become ap-parent in the next two sections, u will be a descendant of v precisely when the cubeassociated with u is contained within the cube associated with v.We designate the empty sequence ∅ as the root of the tree T . The sequence〈i1, . . . , in〉 should be thought of as the vertex in T that is the (in + 1)th child ofthe (in−1 + 1)th child,. . ., of the (i1 + 1)th child of the root. All unordered pairs ofthe form (〈i1, . . . , in−1〉, 〈i1, . . . , in−1, in〉) describe the edges of the tree T . We saythat the edge originates at the vertex 〈i1, . . . , in−1〉 and that it terminates at the41vertex 〈i1, . . . , in−1, in〉. Note that every vertex in the tree that is not the root isuniquely identified by the edge terminating at that vertex. Consequently, given anedge e ∈ E , we define v(e) to be the vertex in V at which e terminates. The vertex〈i1, . . . , in〉 ∈ T also prescribes a unique path, or ray, from the root to this vertex:∅ → 〈i1〉 → 〈i1, i2〉 → · · · → 〈i1, i2, . . . , in〉.We let ∂T denote the collection of all rays in T of maximal (possibly infinite) length.For a fixed vertex v = 〈i1, . . . , im〉 ∈ T , we also define the subtree (of T ) generated bythe vertex v to be the maximal subtree of T with v as the root, i.e. it is the subtree{〈i1, . . . , im, j1, . . . , jk〉 ∈ T : k ≥ 0}.The height of the tree is taken to be the supremum of the lengths of all the se-quences in the tree. Further, we define the height h(·), or level, of a vertex 〈i1, . . . , in〉in the tree to be n, the length of its identifying sequence. All vertices of height n aresaid to be members of the nth generation of the root, or interchangeably, of the tree.More explicitly, a member vertex of the nth generation has exactly n edges joiningit to the root. The height of the root is always taken to be zero.If T is a tree and n ∈ Z+, we write the truncation of T to its first n levels asTn = {〈i1, . . . , ik〉 ∈ T : 0 ≤ k ≤ n}. This subtree is a tree of height at most n.A tree is called locally finite if its truncation to every level is finite, i.e. consists offinitely many vertices. All of our trees will have this property. In the remainder ofthis document, when we speak of a tree we will always mean a locally finite, rootedlabelled tree.Roughly speaking, two trees are isomorphic if they have the same collection ofrays. To make this precise we define a special kind of map between trees; thisdefinition will be very important for us later.Definition 3.1. Let T and T ′ be two trees with equal (possibly infinite) heights. Amap σ : T → T ′ is called sticky if• for all v ∈ T , h(v) = h(σ(v)), and42• u ⊂ v implies σ(u) ⊂ σ(v) for all u, v ∈ T .We often say that σ is sticky if it preserves heights and lineages.A one-to-one and onto sticky map between two trees, when it exists, is said to bean isomorphism and the two trees are said to be isomorphic. Two isomorphic treescan and will be treated as essentially identical objects.Comparing this definition of stickiness with the one described in Section 1.5, thetwo notions do not immediately appear compatible. Indeed, the notion of stickinessdescribed here in Definition 3.1 is, in some ways, far more restrictive than the ge-ometric one discussed in Section 1.5. We will return to these observations shortly,in Section 3.4, after we have established how to encode Euclidean sets by trees ingeneral.3.2 Encoding bounded subsets of the unit intervalby treesThe language of rooted labelled trees is especially convenient for representing boundedsets in Euclidean spaces. This connection is well-studied in the literature [36].We start with [0, 1) ⊂ R. Fix any positive integer M ≥ 2. We define an M -adicrational as a number of the form i/Mk for some i ∈ Z, k ∈ Z+, and an M -adicinterval as [i ·M−k, (i+ 1) ·M−k). For any nonnegative integer i and positive integerk such that i < Mk, there exists a unique representationi = i1Mk−1 + i2Mk−2 + · · · + ik−1M + ik, (3.2)where the integers i1, . . . , ik take values in ZM := {0, 1, . . . ,M − 1}. These integersshould be thought of as the “digits” of i with respect to its base M expansion.An easy consequence of (3.2) is that there is a one-to-one and onto correspondencebetween M -adic rationals in [0, 1) of the form i/Mk and finite integer sequences〈i1, . . . , ik〉 of length k with ij ∈ ZM for each j. Naturally then, we define the tree of43infinite heightT ([0, 1);M) = {〈i1, . . . , ik〉 : k ≥ 0, ij ∈ ZM}. (3.3)The tree thus defined depends of course on the base M ; however, once the base Mhas been fixed, we will omit its usage in our notation, denoting the tree T ([0, 1);M)by T ([0, 1)) instead.Identifying the root of the tree defined in (3.3) with the interval [0, 1) and thevertex 〈i1, . . . , ik〉 with the interval [i ·M−k, (i + 1) ·M−k), where i and 〈i1, . . . , ik〉are related by (3.2), we observe that the vertices of T ([0, 1);M) at height k yield apartition of [0, 1) into M -adic subintervals of length M−k. This tree has a self-similarstructure: every vertex of T ([0, 1);M) has M children and the subtree generated byany vertex as the root is isomorphic to T ([0, 1);M). In the sequel, we will refer tosuch a tree as a full M-adic tree.Any x ∈ [0, 1) can be realized as the intersection of a nested sequence of M -adicintervals, namely{x} =∞⋂k=0Ik(x),where Ik(x) = [ik(x) ·M−k, (ik(x)+1) ·M−k) is the unique M -adic interval in the kthM -adic partition of [0, 1) containing the point x. The point x should be visualizedas the destination of the infinite ray∅ → 〈i1(x)〉 → 〈i1(x), i2(x)〉 → · · · → 〈i1(x), i2(x), . . . , ik(x)〉 → · · ·in T ([0, 1);M). Conversely, every infinite ray∅ → 〈i1〉 → 〈i1, i2〉 → 〈i1, i2, i3〉 · · ·identifies a unique x ∈ [0, 1) given by the convergent sumx =∞∑j=1ijM j.44Thus the tree T ([0, 1);M) can be identified with the interval [0, 1) exactly. Anysubset E ⊆ [0, 1) is then given by a subtree T (E;M) of T ([0, 1);M) consisting of allinfinite rays that identify some x ∈ E. As before, we will drop the notation for thebase M in T (E;M) once this base has been fixed.Any truncation of T (E;M), say up to height k, will be denoted by Tk(E;M) andshould be visualized as a covering of E by M -adic intervals of length M−k. Moreprecisely, 〈i1, . . . , ik〉 ∈ Tk(E;M) if and only if E ∩ [i ·M−k, (i+ 1) ·M−k) 6= ∅, wherei and 〈i1, . . . , ik〉 are related via (3.2).3.3 Encoding higher dimensional bounded subsetsof Euclidean space by treesThe approach to encoding a bounded subset of Euclidean space by a tree extendsreadily to higher dimensions. For any i = 〈j1, . . . , jd〉 ∈ Zd such that i ·M−k ∈ [0, 1)d,we can apply (3.2) to each component of i to obtainiMk= i1M+ i2M2+ · · · + ikMk,with ij ∈ ZdM for all j. As before, we identify i with 〈i1, . . . , ik〉.Let φ : ZdM → {0, 1, . . . ,Md − 1} be an enumeration of ZdM . Define the fullMd-adic treeT ([0, 1)d;M,φ) ={〈φ(i1), . . . , φ(ik)〉 : k ≥ 0, ij ∈ ZdM}. (3.4)The collection of kth generation vertices of this tree may be thought of as the d-foldCartesian product of the kth generation vertices of T ([0, 1);M). For our purposes, itwill suffice to fix φ to be the lexicographic ordering, and so we will omit the notationfor φ in (3.4), writing simply, and with a slight abuse of notation,T ([0, 1)d;M) ={〈i1, . . . , ik〉 : k ≥ 0, ij ∈ ZdM}. (3.5)45As before, we will refer to the tree in (3.5) by the notation T ([0, 1)d) once the baseM has been fixed.By a direct generalization of our one-dimensional results, each vertex 〈i1, . . . , ik〉of T ([0, 1)d;M) at height k represents the unique M -adic cube in [0, 1)d of sidelengthM−k, containing i ·M−k, of the form[j1Mk, j1 + 1Mk)× · · · ×[jdMk, jd + 1Mk).As in the one-dimensional setting, any x ∈ [0, 1)d can be realized as the intersectionof a nested sequence of M -adic cubes. Thus, we view the tree in (3.5) as an encodingof the set [0, 1)d with respect to base M . As before, any subset E ⊆ [0, 1)d thencorresponds to a subtree of T ([0, 1)d;M).3.4 Stickiness as a kind of mapping between treesThroughout this document, we will be concerned with studying certain mappingsbetween pairs of given trees of equal height and comparable base. The first tree Twill represent a set in Euclidean space that roots a collection of thin tubes; i.e. eachpoint in the set will identify the location in Euclidean space of one particular tubein this collection. The second tree S, to which we will map, will encode a set ofdirections or slopes, one for each tube in our collection.Let σ : T → S be a height-preserving transformation that maps full-length raysin T into full-length rays in S, and letKσ :=⋃t∈TPt,σ,where Pt,σ is a thin tube rooted at t and oriented in the direction σ(t). If σ is stickyin the sense of Definition 3.1, then the collection of tubes Kσ is, on average, sticky inthe geometric sense of the term discussed in Section 1.5. This will be made preciselater in Section 5.1, but for now it suffices to imagine the following heuristic.46Fix a pair of trees T and S with the same height H and base M . Consider a pairof tubes rooted at t1, t2 ∈ T respectively. By our work in the previous two sections,we know that |t1−t2| .M−h(u), where u = D(t1, t2) is the youngest common ancestorof t1 and t2 in T .Suppose σ : T → S is sticky in the sense of Definition 3.1. Then since stickymappings preserve heights and lineages, we must have that h(w) ≥ h(u) wherew = D(σ(t1), σ(t2)). Naturally then, we have |σ(t1)− σ(t2)| ≤M−h(w) ≤M−h(u). Inthis way, we see that if in fact |t1− t2| ∼M−h(u), then the separation in the slopes ofPt1,σ and Pt2,σ can be no greater than the separation in their roots. Geometrically,what this means is that if, say, |t1 − t2| ∼ M−H , and if the tubes Pt1,σ, Pt2,σ are tointersect, then that intersection must occur about a unit distance from their roots.Such a collection of tubes exhibits the geometric quality of stickiness illustrated inFigure 1.6.It turns out that this notion of stickiness as a mapping between trees is sufficientto construct Kakeya-type sets over Cantor sets of directions, both in the plane [4]and in general dimensions [31]; see Chapters 5 and 6–11. However, this notion is abit too restrictive when our set of directions is chosen to simply be sublacunary.To see why, note that it is often the case that two points may lie close together inEuclidean space, while their M-adic separation as members of a tree is quite large.To take a concrete example, consider the set D of dyadic rationals on [0, 1] and itscorresponding tree encoding S = S(D; 2). Fix ε > 0. Since D is dense in [0, 1], wemay find d1, d2 ∈ D such that d1 ∈ (12 − ε,12) and d2 ∈ (12 ,12 + ε). Then we see thatthe dyadic separation between d1 and d2 is 2−h(D(d1,d2)) = 1, while their Euclideanseparation is |d1 − d2| ≤ 2ε.Now consider a sticky mapping σ of a collection of root vertices T = T ([0, 12) ∪{1}; 2) into the slope tree S. Suppose |t1 − t2| ∼ ε. Then it is impossible to haveσ(t1) = d1, σ(t2) = d2, else we would require h(D(d1, d2)) ≥ h(D(t1, t2)). Buth(D(t1, t2)) ≥ 1, so this is not possible by the definition of d1 and d2. Thus, anysticky mapping between T and S cannot make the pair of tubes Pt1,σ, Pt2,σ sticky inthe sense of Figure 1.6. Of course, the sticky slope assignment σ may generate othersticky pairs of tubes, but the fact remains that we are undoubtedly neglecting many47other pairs. Clearly, such an assignment is not optimal, in the sense of exploitingthe geometry of stickiness.The notion of tree stickiness is sufficient to establish the theorem of Batemanand Katz (see Chapter 5), as well as our Theorem 1.2. However, once we turn ourattention to Theorem 1.3, we will require a more flexible notion that takes fulleradvantage of the geometric idea of sticky tubes. This will motivate us to definethe notion of a weakly sticky mapping between trees (see Definition 13.6). Sucha mapping will turn out to sufficiently capture the geometric essence of stickinessfor the purposes of constructing Kakeya-type sets over generic sublacunary sets ofdirections.3.5 The tree structure of Cantor-type and subla-cunary setsWe now state and prove a key structural result about our sets of interest for Theo-rem 1.2, the generalized Cantor sets CM .Proposition 3.2. Fix any integer M ≥ 3. Define CM as in Section 1.1. ThenT (CM ;M) ∼= T ([0, 1); 2).That is, the M-adic tree representation of CM is isomorphic to the full binary tree,illustrated in Figure 3.1.Proof. Denote T = T (CM ;M) and T ′ = T ([0, 1); 2). We must construct a bijectivesticky map ψ : T → T ′. First, define ψ(v0) = v′0, where v0 is the root of T and v′0 isthe root of T ′.Now, for any k ≥ 1, consider the vertex 〈i1, i2, . . . , ik〉 ∈ T . We know thatij ∈ ZM for all j. Furthermore, for any fixed j, this vertex corresponds to a kthlevel subinterval of C [k]M . Every such k-th level interval is replaced by exactly twoarbitrary (k + 1)-th level subintervals in the construction of C[k+1]M . Therefore, there48Figure 3.1: A pictorial depiction of the isomorphism between a standard middle-thirds Cantor set and its representation as a full binary subtree of the full baseM = 3 tree.exists N1 := N1(〈i1, . . . , ik〉), N2 := N2(〈i1, . . . , ik〉) ∈ ZM , with N1 < N2, such that〈i1, . . . , ik, ik+1〉 ∈ T if and only if ik+1 = N1 or N2. Consequently, we defineψ(〈i1, i2, . . . , ik〉) = 〈l1, l2, . . . , lk〉 ∈ T ′, (3.6)wherelj+1 =0 if ij+1 = N1(〈i1, . . . , ij〉),1 if ij+1 = N2(〈i1, . . . , ij〉).The mapping ψ is injective by construction and surjectivity follows from the binaryselection of subintervals at each stage in the construction of CM . Moreover, ψ issticky by (3.6).The tree structure of a Cantor-type set is easy to quantify via the above iso-morphism. However, as will soon see, the key properties of this structure are notdependent on the heights of the vertices in the tree, but rather upon the lineagesof those vertices. Imagine we consider some tree T , possibly infinite, and defineD∗ ⊆ T to be the collection of all splitting vertices of T . Then if every vertex v ∈ Tof height J for some J > 0 contains at least N splitting vertices along its lineage,49i.e., #({d ∈ D∗ : v ⊂ d}) ≥ N , the tree TJ has essentially as rich of structure as thefull binary tree of height N . Heuristically, sublacunary sets exhibit this feature forany choice of N > 0. This idea will be rigorously developed in terms of a quantitycalled the splitting number of a tree in the next section. It plays a critical role inboth the proof of our Theorem 1.3 and Bateman’s planar analogue [3].3.6 The splitting number of a treeThere are many ways to quantify the “size” or “spread” of a tree (see [36]). Ofthese, the concept of a splitting number proved to be the most relevant in the planarcharacterization of directions that admit Kakeya-type sets [3]. Not surprisingly, itwill turn out to be equally important for us. As we will see in Section 3.7, this notionactually provides a way of encoding the lacunarity, or lack thereof, of a Euclidean ordirection set. As such, its importance in establishing the results of Section 1.1 willprove critical.We say that a vertex v ∈ T splits in T if it has at least two children in T . Whenit is clear to which tree we are referring, we will just say that v splits, and we willcall v a splitting vertex. Define splitT (R), the splitting number of a ray R in T tobe the number of splitting vertices in T along that ray. The splitting number of avertex v with respect to a tree T is defined to besplitT (v) := maxSv⊆TminRv∈∂SvsplitSv(Rv), (3.7)where the maximum is taken over all subtrees Sv ⊆ T rooted at v, and the minimumis taken over all rays Rv in Sv that originate at the vertex v. Finally, the splittingnumber of the tree T is defined assplit(T ) := maxv∈TsplitT (v). (3.8)To take an easy example, consider the set Ω = {2−j : j ≥ 1}. Then, as is clearfrom Figure 3.2 below, we can quickly observe that split(T (Ω; 2)) = 1. Similarly,50if we consider Ωm = { k2m : 0 ≤ k < 2m}, then we have split(T (Ωm; 2)) = m (seeFigure 3.2). Consequently, the tree depicting all dyadic rationals must have infinitesplitting number, at least when encoded using a base of 2. In fact, this claim holdstrue regardless of the base M chosen to encode the tree. Indeed, consider Ωκm forany integer κ ≥ 1. Then split(T (Ωκm; 2)) = κm, while split(T (Ωκm; 2κ)) = m.Thus, the tree depicting all dyadic rationals with base M = 2κ must have infinitesplitting number. For any base M 6= 2κ for some integer κ ≥ 1, note that Mmust still be on the order of some 2κ˜. Consequently, every M -adic interval on [0, 1)encoded by each vertex of T (Ωm;M) at height j contains and is contained in a fixednumber of 2κ˜-adic intervals of length 2−κ˜j, independent of the height j. This impliessplit(T (Ωm;M)) ∼ split(T (Ωm; 2κ˜)).T3(Ω3; 2)T3(Ω; 2)Figure 3.2: A diagram of the trees representing the Euclidean sets Ω = {2−j : j ≥ 1}and Ωm = { k2m : 0 ≤ k < 2m} for m = 3, up to height 3. Notice that both treesexhibit a kind of self-similar structure, making the calculation of their respectivesplitting numbers particularly easy. The tree encoding Ω is self-similar with respectto the full subtree rooted at any one of the vertices on the leftmost ray. The treeencoding Ωm is self-similar with respect to the full subtree rooted at any vertex ofthe tree.513.6.1 Preliminary facts about splitting numbersThe calculation of the splitting number of a tree was not particularly difficult for thetwo special examples above. However, given an arbitrary tree, it is not immediatelyobvious how to best go about calculating its corresponding splitting number. Thefollowing lemmas will provide the necessary facts to simplify this calculation.Lemma 3.3. Let u, v ∈ T with u ⊆ v. Then splitT (u) ≤ splitT (v).Proof. Let Su be a subtree of T rooted at u. Define Sv→u to be the union of the treeSu with the path in T connecting v to u. This is a subtree of T rooted at v. Since vdoes not split in Sv→u and there are no splitting vertices in Sv→u between v and u,we find that for any ray R in Su,splitSu(R) = splitSv→u(Rv), (3.9)where Rv is the ray in Sv→u rooted at v obtained by extending R to v. Conversely, ifRv is a ray in Sv→u, then (3.9) holds for R = Rv ∩ Su. Maximizing over all subtreesS ⊆ T rooted at u, we have thatsplitT (u) = maxSu⊆TminR∈∂SusplitSu(R)= maxSv→u⊆TminRv∈∂Sv→usplitSv→u(Rv)≤ splitT (v).The last inequality is a consequence of (3.7), since the class of subtrees of the formSv→u is a subcollection of trees rooted at v.Lemma 3.3 says that splitting numbers (of vertices) are monotone nonincreasingin lineages. An immediate consequence of this fact is that split(T ) = splitT (v0),where v0 is the root of T . Our next result says that splitting numbers of trees arealso monotonic in an appropriate sense.Lemma 3.4. Let S ⊆ T . Then split(S) ≤ split(T ).52Proof. By Lemma 3.3, split(S) = splitS(v0), where v0 is the root of S. Since v0 ∈S ⊆ T and any subtree of S is also a subtree of T , we find thatsplitS(v0) = maxSv0⊆SminRv0∈∂Sv0splitSv0 (Rv0)≤ maxSv0⊆TminRv0∈∂Sv0splitSv0 (Rv0)≤ splitT (v0)≤ split(T ),where the last two inequalities are implied by (3.7) and (3.8) respectively. Lemma 3.4follows.A feature of trees with finite splitting number, originally observed in [3, Lemma5], is that all vertices with largest split occur along a single ray. This specialized raywill turn out to be critical in the detection of lacunary limits.Lemma 3.5. Let T be a tree with split(T ) = N . Then there exists a ray R in T (offinite or infinite length) such that a vertex v lies on R if and only if splitT (v) = N ,provided the latter collection contains more than one element.Proof. We prove by contradiction. Suppose there are two vertices u, v ∈ T withsplitT (u) = splitT (v) = N , u 6⊆ v, v 6⊆ u. Then their youngest common ancestorD(u, v) is neither u nor v. By Lemma 3.3, we know that splitT (D(u, v)) ≥ N . Sinceu 6= v, the vertex D(u, v) is actually a splitting vertex. Therefore, splitT (D(u, v)) ≥N + 1. But this contradicts the requirement that split(T ) = N , establishing ourclaim.3.6.2 A reformulation of Theorem 1.3The dichotomy between trees with finite versus infinite splitting number will proveto be our main distinction of interest. Roughly speaking, a tree that has infinitesplitting number in some coordinate system must encode a “large” subset of Eu-clidean space, the threshold of size being determined by sublacunarity. However,53the splitting number of a tree encoding a set is sensitive to the coordinates used torepresent the set. For example, let U and V be the sets constructed in Section 2.1.2.Then split(T (U × V ); 2) = 2, while split(T (ϕ(U × V ); 2)) = ∞ for the coordinatetransformation ϕ(u, v) = (u+v, u−v). More strongly, notice that even the finitenessof the splitting number could be affected by the choice. This consideration featuresprominently in the following proposition which, when taken together with Proposi-tion 3.7, furnishes a restatement of Theorem 1.3 that we will exploit in subsequentchapters.Our proof of Theorem 1.3 will follow a two-step route.Proposition 3.6. Fix a dimension d ≥ 2 and an integer M ≥ 2. If a direction setΩ ⊆ Rd+1 \ {0} is sublacunary (in the sense of Definition 2.7), thensupVsupWΩsupϕsplit(T (ϕ(WΩ);M)) = ∞. (3.10)Here V ranges over the collection of all hyperplanes at unit distance from the origin.For a fixed V, the set WΩ ranges over all relatively compact subsets of CΩ ∩ V, andthe innermost supremum is taken over all coordinate choices ϕ = (a,B) on V, wherea ∈ V is the point closest to the origin and B = {v1, · · · ,vd} is any orthonormalbasis of V− a. In other words, ϕ represents a rotation in V centred at a, withϕ(CΩ ∩ V) ={(x1, · · · , xd) : x = a +d∑j=1xjvj ∈ CΩ ∩ V}.Thus for every N ≥ 1, there exists a hyperplane VN , a relatively compact subsetWN of CΩ ∩ VN , and a coordinate system ϕN on VN such thatsplit(T (ϕN(WN);M)) > N. (3.11)Proposition 3.7. If a direction set Ω obeys (3.10) for some M ≥ 2, then Ω admitsKakeya-type sets.Proposition 3.7 will be the subject of Chapters 15 – 19. We prove Proposition 3.654in Section 3.7 below.3.7 Lacunarity on treesA distinctive feature in the planar characterization of Kakeya-type sets [3] is theobservation that the lacunarity of a set is reflected in the structure of its tree. Fol-lowing the ideas developed there, we recast the concept of finite order lacunarity ofa one-dimensional set using the structure of the splitting vertices of its tree. Thisprovides a tool of convenience in the proof of Proposition 3.6, the main objective ofthis section.Lemma 3.8. For any M ≥ 2, N ≥ 1, there is a constant C = C(N,M) with thefollowing property. If a relatively compact set U ⊆ R is such that split(T (U ;M)) =N , then U can be covered by the C-fold union of sets in Λ(N,M−1) as described inDefinition 2.2.The proof of this lemma will be presented later in this section. Assuming this,the proof of the proposition is completed as follows.Proof of Proposition 3.6. We prove the contrapositive, starting with the assumptionthatsupVsupWΩsupϕsplit(T (ϕ(WΩ;M)) = N <∞. (3.12)Fix an arbitrary coordinate system ϕ = (a,B) of V and let pij denote the projectionmaps defined in (2.2) with respect to this choice. For the remainder of this proof, wewill assume that V is represented in these coordinates, so that pij may be thoughtof as the coordinate projections. Let W = WΩ be an arbitrary relatively compactsubset of CΩ ∩ V. Since the tree encoding a set matches that of its closure, we maysuppose without loss of generality that W = WΩ is compact in V.For any 1 ≤ j ≤ d + 1, we create a subset Wj ⊆ W that contains for everyxj ∈ pij(W ) a unique point x ∈ W with pij(x) = xj. For concreteness, x could bechosen to be minimal in pi−1j (xj) ∩W with respect to the lexicographic ordering. In55other words, pij restricted to Wj is a bijection onto pij(W ). We claim thatsplit(T (pij(W );M)) ≤ split(T (Wj;M)). (3.13)Assuming this for the moment, we obtain from the hypothesis (3.12) and Lemma3.4 that split(T (pij(W );M) ≤ split(T (W ;M)) ≤ N . Applying Lemma 3.8 to U =pij(W ), we see that there is a constant C (uniform in V, ϕ, j and W ) such that theprojections pij(W ) can be covered by the C-fold union of one-dimensional lacunarysets of order ≤ N and lacunarity ≤ M−1. Thus, W = WΩ is admissible lacunary oforder at most N according to Definition 2.5. Hence Ω is admissible lacunary of finiteorder as a direction set by Definition 2.7.It remains to establish (3.13). Any infinite ray R = R(xj) in ∂T (pij(W );M)corresponds to a point xj ∈ pij(W ). Let R∗ = R∗(x) ∈ ∂T (Wj;M) denote the raythat represents pi−1j (xj) = x. This establishes a bijection between the collection ofrays in the two trees. Let v0 and v∗0 denote the roots of the trees T (pij(W );M) andT (W ;M) respectively, so that pij(v∗0) = v0. If S is a subtree of T (pij(W );M) rootedat v0, let us denote by S∗ the subtree of T (Wj;M) rooted at v∗0 generated by all raysR∗ such that R is a ray of S. It is clear that if a vertex v on R(xj) splits in S, thenthere are two points xj 6= x′j in pij(W ) lying in distinct children of v. This impliesthat x = pi−1j (xj) and x′ = pi−1j (x′j) lie in distinct children of v∗, which denotes thevertex of height h(v) on R∗(x). This makes v∗ a splitting vertex of S∗. Thus everysplitting vertex of S lying on R generates a splitting vertex of S∗ lying on R∗ at thesame height. As a result, splitS(R) ≤ splitS∗(R∗). Combining these facts with thedefinition of the splitting number of a tree, we obtainsplit(T (pij(W );M)) = maxSminR∈∂SsplitS(R)≤ maxS∗minR∗∈∂S∗splitS∗(R∗)≤ split(T (Wj;M)).In view of Lemma 3.3, the maxima in the first and second lines above are taken overall subtrees S and S∗ rooted at v0 and v∗0 respectively. This completes the proof of56(3.13) and hence of Proposition 3.6.We now turn to the proof of the lemma on which the argument above was pred-icated.Proof of Lemma 3.8. We apply induction on N . The base case N = 1 will be treatedmomentarily in Lemma 3.9. Proceeding to the induction step, let R∗ denote aninfinite ray of the tree T = T (U ;M) that contains all the vertices {v∗ : splitT (v∗) =N}. The existence of such a ray has been established in Lemma 3.5. For everyvertex v in T (U ;M) which does not lie on R∗ but whose parent does, we definea set Uv as follows: Tv = T (Uv;M), where Tv denotes the maximal subtree of Trooted at v. The definition of the ray R∗ dictates that each Uv has the propertythat split(T (Uv;M)) ≤ N − 1. By the induction hypothesis, there exists a constantC = C(N − 1,M) such that each Uv is covered by the C-fold union of sets inΛ(N − 1;M−1). The set U can therefore be covered by the C-fold union of sets U [i],where each U [i] shares a tree structure similar to U : it contains the point identifiedby R∗, with the additional feature that now U [i]v ∈ Λ(N − 1;M−1) for every v ∈ V [i],whereV [i] :={v ∈ T (U [i];M) : v /∈ R∗ but parent of v is in R∗}.For every vertex v ∈ V [i], let av denote the left hand endpoint of the M -adicinterval represented by v. The tree encoding the collection of points A = {av : v ∈V [i]} contains the ray R∗; indeed the only splitting vertices of T (A;M) lie on R∗.Therefore split(T (A;M)) = 1. Hence, by Lemma 3.9, A is at most a C-fold unionof monotone lacunary sequences with lacunarity M−1, each converging to the pointidentifying R∗. Let us continue to denote by A one such monotone (say decreasing)sequence. If a = av and b are two successive elements of this sequence with a < b,then U [i] ∩ [a, b) = U [i]v , which is in Λ(N − 1;M−1). Thus U [i] is in Λ(N ;M−1)according to Definition 2.2, completing the proof.Lemma 3.9. Fix M ≥ 2, and let A ⊆ R be a relatively compact set with the propertythat split(T (A;M)) = 1. Then A can be written as the union of at most 6M lacunarysequences (defined in Definition 2.1) each with lacunarity constant ≤M−1.57Proof. The argument here closely follows the line of reasoning in [3, Remark 2, page60]. By Lemma 3.5, there is a ray R∗ in T (A;M) of infinite length such that all thesplitting vertices of T (A;M) lie on it. The ray R∗ uniquely identifies a point in R,say a∗ = α(R∗). Any ray that is not R∗ but is rooted at a vertex of R∗ is thereforenon-splitting. Thus for every j = 0, 1, 2, · · · there exists at most M − 1 rays Rj inT (A;M) whose M -adic distance from R∗ is j. In other words, if aj = α(Rj) is thepoint in A identified by Rj, then there are at most M − 1 distinct points aj 6= a∗such thath(D(a∗, aj)) = h(D(α(R∗), α(Rj))) = j. (3.14)We define two subsets A± of A, containing respectively points a ≥ a∗ and a ≤ a∗.This decomposes T (A;M) into two subtrees T (A±;M). Let us focus on T (A+;M),the treatment for the other tree being identical. We decompose T (A+;M) as theunion of at most M trees T (Ai+;M), i ∈ ZM , constructed as follows. The treeT (Ai+;M) contains the ray R∗, and for every vertex v in R∗ the ray in T (A+;M),if any, descended from the ith child of v. In view of the discussion in the precedingparagraph, if there exists an integer j for which a ray Rj in T (Ai+;M) obeys (3.14),then such a ray must be unique.We now fix i ∈ ZM and proceed to cover Ai+ by a threefold union of lacunarysequences converging to a∗. Let {n1 < n2 < · · · } be the subsequence of integers withthe property that Rj ∈ T (Ai+;M) if and only if j = nk for some k. The importantobservation is that if nk+2 is a member of this subsequence, thenank − a∗ ≥1Mnk+2. (3.15)We will return to the proof of this statement in a moment, but a consequence of itand (3.14) is that for any k ≥ 0 and fixed ` = 0, 1, 2,an3(k+1)+` − a∗ ≤M−n3k+3+` = M−n3k+3+`+n3k+2+`M−n3k+2+` ≤M−1(an3k+` − a∗).Thus for every fixed ` = 0, 1, 2, the sequence A` = {an3k+` : k ≥ 0} is covered by alacunary sequence with constant ≤M−1 converging to a∗. Since Ai+ is the union of58{A` : ` = 0, 1, 2}, the result follows.It remains to settle (3.15), which is best explained by Figure 3.3. If Ij is the M -a∗ ank+2 ank+1 ank|J | = M−nk+2ank − a∗Ik+2Ik+1 IkFigure 3.3: A figure explaining inequality (3.15) when M = 2 and nk = k.adic interval of length M−nj containing a∗, then Ik+2 cannot share a right endpointwith Ik+1, since this would prevent the existence of a point ank+1 ≥ a∗ obeying (3.14)with j = nk+1. Thus a∗ (in Ik+2) and ank (which is to the right of Ik+1) must lie onopposite sides of J , the rightmost M -adic subinterval of length M−nk+2 in Ik+1. Thisimplies ank − a∗ ≥ |J |, which is the conclusion of (3.15).59Chapter 4Electrical circuits and percolationon treesAs mentioned in Sections 1.4 and 1.5, a key component in the work of Bateman andKatz [4], [3], as well as in the work that forms the backbone of this document [31],[32], is a rather famous estimate from the theory of percolation processes on trees.The literature on this and related percolation items is massive; e.g. see [20], [36].We will discuss only a very small piece of this vast topic, sufficient for our purposes.4.1 The percolation process associated to a treeThe special probabilistic process of interest to us is called a bond percolation ontrees. Imagine a liquid that is poured on top of some porous material. How will theliquid flow - or percolate - through the holes of the material? How likely is it thatthe liquid will flow from hole to hole in at least one uninterrupted path all the wayto the bottom? The first question forms the intuition behind a formal percolationprocess, whereas the second question turns out to be of critical importance to theproof of Theorems 1.2 and 1.3.Although it is possible to speak of percolation processes in far more general terms(see [20]), we will only be concerned with a percolation process on a tree. Accordingly,60given some tree T with vertex set V and edge set E , we define an edge-dependentBernoulli (bond) percolation process to be any collection of random variables {Xe :e ∈ E}, where Xe is Bernoulli(pe) with pe < 1. The parameter pe is called thesurvival probability of the edge e. We will always be concerned with a particular typeof percolation on our trees: we define a standard Bernoulli(p) percolation to be onewhere the random variables {Xe : e ∈ E} are mutually independent and identicallydistributed Bernoulli(p) random variables, for some p < 1. In fact, for our purposes,it will suffice to consider only standard Bernoulli(12) percolations.Rather than imagining a tree with a percolation process as the behaviour of aliquid acted upon by gravity in a porous material, it will be useful to think of thepercolation process as acting more directly on the mathematical object of the treeitself. Given some percolation process on a tree T , we will think of the event {Xe = 0}as the event that we remove the edge e from the edge set E , and the event {Xe = 1}as the event that we retain this edge; denote the random set of retained edges byE∗. Notice that with this interpretation, after percolation there is no guarantee thatE∗, the subset of edges that remain after percolation, defines a subtree of T . In fact,it can be quite likely that the subgraph that remains after percolation is a union ofmany disconnected subgraphs of T .For a given edge e ∈ E , we think of p = Pr(Xe = 1) as the probability that weretain this edge after percolation. The probability that at least one uninterruptedpath remains from the root of the tree to its bottommost level is given by the sur-vival probability of the corresponding percolation process. More explicitly, given apercolation on a tree T , the survival probability after percolation is the probabilitythat the random variables associated to all edges of at least one ray in T take thevalue 1; i.e.Pr (survival after percolation on T ) := Pr(⋃R∈∂T⋂e∈E∩R{Xe = 1}). (4.1)Estimation of this probability is what will be utilized in the proofs of Theorems 1.2and 1.3. This estimation will require reimagining a tree as an electrical network.614.2 Trees as electrical networksFormally, an electrical network is a particular kind of weighted graph. The weightsof the edges are called conductances and their reciprocals are called resistances. Inhis seminal works on the subject, Lyons visualizes percolation on a tree as a certainelectrical network. In [34], he lays the groundwork for this correspondence. Whilehis results hold in great generality, we describe his results in the context of standardBernoulli percolation on a locally finite, rooted labelled tree only.A percolation process on the truncation of any given tree T is naturally associatedto a particular electrical network. To see this, we truncate the tree T at height Nand place the positive node of a battery at the root of TN . Then, for every rayin ∂TN , there is a unique terminating vertex; we connect each of these vertices tothe negative node of the battery. A resistor is placed on every edge e of TN withresistance Re defined by1Re= 11 − pe∏∅⊃v(e′)⊇v(e)pe′ . (4.2)Notice that the resistance for the edge e is essentially the reciprocal of the probabilitythat a path remains from the root of the tree to the vertex v(e) after percolation.For standard Bernoulli(12) percolation, we haveRe = 2h(v(e))−1. (4.3)One fact that will prove useful for us later is that connecting any two vertices ata given height by an ideal conductor (i.e. one with zero resistance) only decreasesthe overall resistance of the circuit. This will allow us to more easily estimate thetotal resistance of a generic tree.Proposition 4.1. Let TN be a truncated tree of height N with corresponding electricalnetwork generated by a standard Bernoulli(12) percolation process. Suppose at heightk < N we connect two vertices by a conductor with zero resistance. Then the resultingelectrical network has a total resistance no greater than that of the original network.Proof. Let u and v be the two vertices at height k that we will connect with an62ideal conductor. Let R1 denote the resistance between u and D(u, v), the youngestcommon ancestor of u and v; let R2 denote the resistance between v and D(u, v).Let R3 denote the total resistance of the subtree of TN generated by the root uand let R4 denote the total resistance of the subtree of TN generated by the root v.These four connections define a subnetwork of our tree, depicted in Figure 4.1(a).The connection of u and v by an ideal conductor, as pictured in Figure 4.1(b), canonly change the total resistance of this subnetwork, as that action leaves all otherconnections unaltered. It therefore suffices to prove that the total resistance of thesubnetwork comprised of the resistors R1, R2, R3 and R4 can only decrease if u andv are joined by an ideal conductor.(a)D(u, v)u v+−R1 R2R3 R4(b)D(u, v)u ∼ v+−R1 R2R3 R4Figure 4.1: (a) The original subnetwork with the resistors R1, R3 and R2, R4 inseries; (b) the new subnetwork obtained by connecting vertices u and v by an idealconductor.In the original subnetwork, the resistors R1 and R3 are in series, as are theresistors R2 and R4. These pairs of resistors are also in parallel with each other.Thus, we calculate the total resistance of this subnetwork, Roriginal:Roriginal =(1R1 +R3+ 1R2 +R4)−163= (R1 +R3)(R2 +R4)R1 +R2 +R3 +R4. (4.4)After connecting vertices u and v by an ideal conductor, the structure of our sub-network is inverted as follows. The resistors R1 and R2 are in parallel, as are theresistors R3 and R4, and these pairs of resistors are also in series with each other.Therefore, we calculate the new total resistance of this subnetwork, Rnew, asRnew =(1R1+ 1R2)−1+(1R3+ 1R4)−1= R1R2(R3 +R4) +R3R4(R1 +R2)(R1 +R2)(R3 +R4). (4.5)We claim that (4.4) is greater than or equal to (4.5). To see this, simply cross-multiply these expressions. After cancellation of common terms, our claim reducestoR21R24 +R22R23 ≥ 2R1R2R3R4.But this is trivially satisfied since (a− b)2 ≥ 0 for any real numbers a and b.The main consequence of this observation that we draw upon in Lemmas 15.4and 11.3 is given by the following corollary.Corollary 4.2. Given a subtree TN of height N contained in the full d-dimensionalM-adic tree, let R(TN) denote the total resistance of the electrical network that corre-sponds to standard Bernoulli(12) percolation on this tree, in the sense of the theoremof Lyons as given in Theorem 4.3. ThenR(TN) ≥N∑k=12k−1nk, (4.6)where nk denote the number of its kth generation vertices in TN .Proof. To show this, we construct an auxiliary electrical network from the one nat-urally associated to our tree TN , as follows. For every k ≥ 1, we connect all verticesat height k by an ideal conductor to make one node Vk. Call this new circuit E.64The resistance of E cannot be greater than the resistance of the original circuit, byProposition 4.1.Fix k, 1 ≤ k ≤ N , and let Rk denote the resistance in E between Vk−1 andVk. The number of edges between Vk−1 and Vk is equal to the number nk of kthgeneration vertices in TN , and each edge is endowed with resistance 2k−1 by (4.3).Since these resistors are in parallel, we obtain1Rk=nk∑112k−1= nk2k−1.This holds for every 1 ≤ k ≤ N . Since the resistors {Rk}Nk=1 are in series, R(TN) ≥R(E) =∑Nk=1Rk, establishing inequality (11.8).4.3 Estimating the survival probability after per-colationWe now present Lyons’ pivotal result linking the total resistance of an electricalnetwork and the survival probability under the associated percolation process.Theorem 4.3 (Lyons, Theorem 2.1 of [35]). Let T be a tree with mutually associatedpercolation process and electrical network, and let R(T ) denote the total resistanceof this network. If the percolation is Bernoulli, then11 +R(T )≤ Pr(T ) ≤ 21 +R(T ),where Pr(T ) denotes the survival probability after percolation on T .We will not require the full strength of this theorem. A reasonable upper boundon the survival probability coupled with the result of Proposition 4.1 will suffice forour applications. The sufficient simpler version of Theorem 4.3 that we state andprove below was essentially formulated by Bateman and Katz [4].65Proposition 4.4. Let M ≥ 2 and let T be a subtree of a full M-adic tree. Let R(T )and Pr(T ) be as in Theorem 4.3. Then under Bernoulli percolation, we havePr(T ) ≤ 21 +R(T ). (4.7)Proof. We will only focus on the case when R(T ) ≥ 1, since otherwise (4.7) holdstrivially. We prove this by induction on the height of the tree N . When N = 0, then(4.7) is trivially satisfied. Now suppose that up to height N − 1, we havePr(T ) ≤ 21 +R(T ).Suppose T is of height N . We can view the tree T as its root together withat most M edges connecting the root to the subtrees T1, . . . , TM of height N − 1generated by the terminating vertices of these edges. If there are k < M edgesoriginating from the root, then we take M − k of these subtrees to be empty. Notethat by the induction hypothesis, (4.7) holds for each Tj. To simplify notation, wedenotePr(Tj) = Pj and R(Tj) = Rj,taking Pj = 0 and Rj = ∞ if Tj is empty.Using independence and recasting Pr(T ) as one minus the probability of notsurviving after percolation on T , we have the formula:Pr(T ) = 1 −M∏k=1(1 − 12Pk).Note that the function F (x1, . . . , xM) = 1 − (1 − x1/2)(1 − x2/2) · · · (1 − xM/2) ismonotone increasing in each variable on [0, 2]M . Now defineQj :=21 +Rj.66Since resistances are nonnegative, we know that Qj ≤ 2 for all j. Therefore,Pr(T ) = F (P1, . . . , PM)≤ F (Q1, . . . , QM)≤ 12M∑k=1Qk.Here, the first inequality follows by monotonicity and the induction hypothesis. Plug-ging in the definition of Qk, we find thatPr(T ) ≤M∑k=111 +Rk.But since each resistor Rj is in parallel, we know that1R(T )=M∑k=111 +Rk.Combining this formula with the previous inequality and recalling that R(T ) ≥ 1,we havePr(T ) ≤ 1R(T )≤ 21 +R(T ),as required.67Chapter 5Kakeya-type sets over a Cantor setof directions in the planeThe construction of Kakeya-type sets over a Cantor set of directions Ω = CM is con-siderably easier than the general case, due to the very particular and fixed structureof the corresponding slope tree, T (CM ;M), as established in Proposition 3.2. Thisstatement holds true regardless of dimension considered, and as such we will beginour discussion in this simplified setting. Starting with the work of Bateman andKatz [4], we will outline their construction in the plane and point out the additionalchallenges that appear in the higher dimensional case. We will proceed into a discus-sion of the added difficulties that arise when the set of directions is generalized to asublacunary set. In Chapter 6, we will begin the details of our general constructionin any dimension. The framework of the analysis in all subsequent chapters will haveits foundation in the arguments of the present chapter.5.1 The result of Bateman and KatzIn [4], Bateman and Katz consider what happens when thin tubes in the planeare assigned directions that arise in a natural way from the standard middle-thirdCantor set on [0, 1), denoted here by C. For every n ≥ 1, they define a random sticky68mappingσn : Tn([0, 1); 3) → Tn(C; 3). (5.1)Recall that such a mapping between trees is set to preserve heights and lineages,per Definition 3.1. Note that as an immediate consequence of Proposition 3.2, wehaveTn(C; 3) ∼= Tn([0, 1); 2) for all n ≥ 1. (5.2)The random mechanism of the mapping σn then assigns one edge in Tn([0, 1); 3) toone edge in Tn(C; 3), independently and with equal probability. The collection of allsticky mappings is thus uniformly distributed on the edge set of Tn([0, 1); 3).Now fix n ≥ 1; for each t ∈ Tn([0, 1); 3), we define a random tube Pt,σn in R2 withprincipal axis given by the line segment from (0, t) to (1, t+σn(t)) and cross-sectionaldiameter 3−(n+1). Here, as in the sequel, we abuse notation slightly and identifyt ∈ Tn([0, 1); 3) and σn(t) ∈ Tn(C; 3) with their naturally associated real numbers on[0, 1), via (3.2) and (3.3). Consequently, we consider the random collection of tubesKσn :=⋃t∈Tn([0,1);3)h(t)=nPt,σn , (5.3)with the slope of each tube a member of the standard middle-third Cantor set at itsnth stage of construction.Bateman and Katz prove the following two estimates on the random sets Kσn :for any sticky map σn, |Kσn | &log nn, (5.4)andthere exists a sticky σn such that∣∣∣∣Kσn ∩([13, 1]× R)∣∣∣∣.1n. (5.5)Notice that these estimates provide precisely the Kakeya-type set condition (1.1) forany n ≥ 1. More precisely, in the notation of Definition 1.1, we set Kσn = E∗n(A0)and Kσn ∩[13 ,23]= En, where A0 = 3. (Note that the tubes Pt,σn just defined in(5.3) are not the same tubes P (n)t referred to in Definition 1.1.) Then the estimates69(5.4) and (5.5) yieldlimn→∞|E∗n(A0)||EN |= ∞,which is the requirement of (1.1). Consequently, we conclude that the set of directionsΩ = {ω ∈ S1 : tanω ∈ C} admits Kakeya-type sets.5.1.1 Proof of inequality (5.4)Estimate (5.4) is proven by combining a general observation about intersections oflike sets with the Co´rdoba estimate (1.5) applied on a proper partition of the randomset Kσn . The estimate (5.5) follows from a probabilistic argument that relies onLyons’ theorem about percolation on trees, (see Theorem 4.3). We will sketch theseproofs in a bit more detail, as they will provide the analytical foundation for themore general cases to follow. Since n is fixed in what follows, to simplify notationwe will write σ in place of σn; we will also write Tn in place of Tn([0, 1); 3) and Cninstead of Tn(C; 3).For the lower bound estimate (5.4), we begin with the following lemma.Lemma 5.1 (Bateman and Katz). Suppose (X,M, µ) is a measure space and A1, . . . , Amare sets with µ(Ai) = α for every i. Let L > 0, and suppose thatm∑j=1m∑i=1µ(Ai ∩ Aj) ≤ L.Thenµ(m⋃i=1Ai)≥ m2α216L.Proof. By the pigeonhole principle, there exists a set M ⊂ {1, . . . ,m} with #(M) ≥m2 such that whenever i ∈M , we havem∑j=1µ(Ai ∩ Aj) ≤2Lm.70For any such i, we rewrite this inequality as1α∫Aim∑j=11Aj(x)dµ(x) ≤2Lmα.Since∑1Aj(x) is a nonnegative function, the only way its average value over Ai canbe bounded by 2Lmα is if there exists a subset Bi ⊆ Ai with µ(Bi) ≥α2 such that forevery x ∈ Bi,m∑j=11Aj(x) ≤4Lmα.Now we divide through by the quantity on the right-hand side and integrate over⋃i∈MBi:mα4L∫⋃i∈M Bim∑j=11Aj(x)dµ(x) ≤ µ(⋃i∈MBi).Remembering that Bi ⊆ Ai for all i ∈M , and interchanging the sum and the integralon the left-hand side, we concludeµ(m⋃i=1Ai)≥ mα4L∑i∈Mµ(Bi) ≥m2α216L,using the trivial bounds µ(Bi) ≥ α2 and #(M) ≥m2 .Proof of (5.4). For 0 ≤ j < 12 log n, we define Pt,σ,j := Pt,σ ∩ [3−j, 3−(j−1)], and notethat |Pt,σ,j| ∼ 3−(j+n). Then the collection {Pt,σ,j}j is disjoint over j. In light ofLemma 5.1, we would like to show that the measure of the union of these pieces issmaller than some appropriate quantity for all j < 12 log n. Consequently, it sufficesto show that∑t1∈Tnh(t1)=n∑t2∈Tnh(t2)=n|Pt1,σ,j ∩ Pt2,σ,j| .n32j. (5.6)The height restrictions h(t1) = h(t2) = n will hold throughout this section, but wewill henceforth suppress this in the notation to save on clutter. Once we have (5.6)71for every 0 ≤ j < 12 log n, using Lemma 5.1, we calculate∣∣∣∣∣⋃t∈TnPt,σ,j∣∣∣∣∣&1nfor each j < 12 log n. Thus,∣∣Kσ ∩ [n−c, 1)∣∣&log nn,which implies (5.4).Notice that the diagonal term of the sum in (5.6) satisfies the proper boundsince j < 12 log n, and that in fact this is the best we can do. Now if t1 6= t2 andPt1,σ,j ∩ Pt2,σ,j 6= ∅, then|σ(t1) − σ(t2)| & 3j|t1 − t2|, (5.7)since θ & sin θ for θ small. Pairing this bound on the difference in the slopes σ(t1)and σ(t2) with the Co´rdoba estimate (1.5), we have|Pt1,σ,j ∩ Pt2,σ,j| .132n+j|t1 − t2|. (5.8)Recall that the notation D(t1, t2) represents the youngest common ancestor of t1and t2 in the tree Tn. Now, the stickiness of σ and (5.7) imply that|ID(t1,t2)| & 3j|t1 − t2|, (5.9)where Iu denotes the unique triadic interval corresponding to the vertex u (see Sec-tion 3.2). We will shortly prove the following useful counting lemma.Lemma 5.2. Let t1, t2 ∈ Tn([0, 1); 3) and set u = D(t1, t2). Suppose |Iu| & 3j|t1−t2|,and define the set Ak,j(u) := {(t1, t2) ∈ D−1(u) : |t1 − t2| ∼ 3−j−k|Iu|}. Then#(Ak,j(u)) . 32n−2j−2k−2h(u).72Note that by definition each pair (t1, t2) belongs to exactly one Ak,j(u). Thus,after plugging in the bound (5.8), we rewrite the off-diagonal part of the sum in (5.6)as∑t1∈Tn∑t2∈Tnt2 6=t1|Pt1,σ,j ∩ Pt2,σ,j| .∑u∈Tn∑(t1,t2)∈D−1(u)3j |t1−t2|.|Iu|132n+j|t1 − t2|.∑u∈Tn∑k≥0∑(t1,t2)∈Ak,j(u)132n+j|t1 − t2|.∑u∈Tn∑k≥0∑(t1,t2)∈Ak,j(u)132n−k|Iu|.Using Lemma 5.2, we complete the estimation as∑u∈Tn∑k≥032n−2j−2k−2h(u)32n−k|Iu|.∑u∈Tn∑k≥03−k|Iu|32j.n32j,where we have used the fact that 3−k is summable, and that the intervals Iu forma partition of the unit interval over all u of fixed height; since Tn has n + 1 distinctheights, (5.6) follows.It remains to prove Lemma 5.2. We will accomplish this by counting the numberof permissible vertices t2 at height n after fixing a vertex t1. Then we will count thenumber of t1.Proof of Lemma 5.2. If (t1, t2) ∈ Ak,j(u), then the lineages of t1 and t2 split at heighth(u), yet remain close enough in the tree Tn([0, 1); 3) so that |t1−t2| ∼ 3−j−k−h(u). Tocount the number of permissible pairs (t1, t2), let ul denote the lth child of u. We willrestrict attention to the pairs (t1, t2) that are descendants of u1 and u2 respectively.Counting over all options will introduce no more than a constant factor of(32)to thefinal count.Fix t1 ⊆ u1. Notice that the only permissible t2 now lie in the intersection of theinterval associated to u2 with an interval of length 3−j−k−h(u) containing t1. Thus,to bound the number of t2 with such a t1 fixed, it suffices to simply 3−n-separate the73interval of length 3−j−k−h(u). Similarly, if we now let t1 vary, t1 can only be chosen tolie no more than a distance of . 3−j−k−h(u) from the triadic interval associated to u2;hence, we again 3−n-separate the interval of length 3−j−k−h(u) to bound the numberof t1. Thus, we have the size estimate #(Ak,j(u)) . 32n−2j−2k−2h(u) as claimed.The lower bound (5.4) holds for any sticky collection of tubes, but notice that theonly time we require the fact that σ is sticky is for the estimate (5.9). Without such anestimate, there would be no restriction placed on the height of the youngest commonancestor of the two points t1 and t2 that generate intersecting tubes. As in ourgeneric discussion in Section 1.5, stickiness is a property that provides a quantitativeconnection between the location of tubes in space and their orientations. By requiringPt1,σ,j ∩ Pt2,σ,j 6= ∅ in the proof above, we are fixing a region in space where thesetubes can overlap. This naturally places a restriction on the permissible slopes thatcan be assigned to any two base points t1 and t2 in order for a nonempty intersectionto occur in the [3−j, 3−(j−1)] × R strip. But it is our enforced notion of stickiness,formulated in the context of a mapping between trees, that provides the necessaryquantitative information to effectively exploit the geometry and establish (5.4). Thiswill prove to be sufficient for the purposes of our Theorem 1.2, but Theorem 1.3 willrequire a more flexible notion of stickiness between root and slope trees to yield tothese methods.5.1.2 Proof of inequality (5.5)In contrast to (5.4), it is quite clear that estimate (5.5) cannot be expected to holdfor an arbitrary sticky σ; for example, consider the sticky map that assigns to everybase point t ∈ Tn the same slope, say 0 ∈ Cn. However, the percolation argumentexploited by Bateman and Katz shows that (5.5) does hold for a typical sticky σ, aswe shall now see.Proof of (5.5). Fix a point (x, y) ∈ (13 , 1)×R. We begin by observing that for everybase point t ∈ Tn, there is at most one slope c(x,y)(t) ∈ Cn such that (x, y) ∈ Pt,c(x,y)(t)(here, we have abused notation slightly and let Pt,c represent the tube with principal74axis given by the line segment (0, t) to (1, t+c) with cross-sectional diameter 3−(n+1)).Indeed, suppose two distinct slopes c1 6= c2 existed such that (x, y) ∈ Pt,c1 ∩ Pt,c2 .Then there must exist a1, a2 ∈ R with |a1|, |a2| ≤ 12 ·3−(n+1), the cross-sectional radiusof a tube, such thaty = t+ a1 + c1x = t+ a2 + c2x.But since |c2− c1| ≥ 3−n by construction, it must follow that x ≤ 13 , a contradiction.Let Poss(x, y) denote the subtree of Tn where t ∈ Poss(x, y) if and only if c(x,y)(t)exists. This is a deterministic object, dependent only on the choice of (x, y). Nownotice that the event (x, y) ∈ Kσ happens only if σ(t) = c(x,y)(t) for some t ∈Possn(x, y); i.e. for some t ∈ Poss(x, y) with h(t) = n. Equivalently, this can onlyhappen if σ(tk) = c(x,y)(tk) for every ancestor tk of t ∈ Possn(x, y), 0 ≤ k ≤ n.Recall the definition of σ as given by (5.1). By the structure of the slope tree givenin (5.2), we see that defining a random sticky map σ is equivalent to defining acollection of 3n+1 − 1 independent Bernoulli(12) random variables {Xe : e ∈ E(Tn)},one corresponding to each edge in the tree Tn. More precisely, by (5.2), and in orderto preserve stickiness, each edge e ∈ E(Tn) can be mapped to only one of two possibleedges in Cn. We declare that e maps to the first of these two possibilities if Xe = 0,and that e maps to the second otherwise.Now, if (x, y) ∈ Kσ, then we have already seen that there must exist a t ∈Possn(x, y) so that σ(tk) = c(x,y)(tk) for every ancestor tk ⊇ t, 0 ≤ k ≤ n. As inSection 4.1, we will equate the event {Xe = 0} with the action of removing the edgee from the edge set E(Tn), and the event {Xe = 1} with the action of retaining thisedge. Thus, we see that the event (x, y) ∈ Kσ occurs only if a path from the rootto the bottommost level remains after percolation on the tree Poss(x, y). LettingPr(Poss(x, y)) denote the probability of this event, Lyons’ Theorem 4.3 tells us thatPr(Poss(x, y)) ≤ 21 +R(Poss(x, y)), (5.10)where R(Poss(x, y)) is the resistance of the electrical network associated to the treePoss(x, y), as defined in Section 4.2.75We claim that R(Poss(x, y)) & n. Notice that the number of vertices at heightk in Poss(x, y) is bounded above by . 2k. Indeed, there are exactly 2k differentvertices at height k in the slope tree Cn; thus, using a similar argument as the onepresented in the first paragraph of this proof, for any fixed c ∈ Cn at height k, thereare a finite number of t ∈ Tn of height k that could possibly yield c(x,y)(t) = c.Now consider the electrical network associated to Poss(x, y). By Proposition 4.1,connecting all vertices in Poss(x, y) at height k with an ideal conductor can onlyreduce the total resistance of the circuit; make this transformation at each heightk > 0 and denote the resulting tree by T (x, y). Thus, there are . 2k resistors, eachwith resistance ∼ 2k, connected in parallel between height k − 1 and height k inT (x, y). This gives a resistance & 1 between heights k − 1 and k, and consequentlya total resistance of the circuit & n. This paired with (5.10) establishes the boundPr((x, y) ∈ Kσ) . 1n .Finally, we use this bound to estimate the expected measure of Kσ ∩ ([13 , 1]×R).Observing that we must have y ∈ [0, 2] if (x, y) ∈ Kσ, we calculateEσ(∣∣∣∣Kσ ∩([13, 1]× R)∣∣∣∣)=∫(∫ 113∫ 201Kσ(x, y)dydx)dσ=∫ 113∫ 20Pr((x, y) ∈ Kσ)dydx.1n.Thus, there is a choice of sticky map σ such that (5.5) holds. Moreover, since therandom variable σ is distributed uniformly over all sticky maps (this is by definitionof our random slope assignment), this argument shows that in fact most sticky σmust satisfy (5.5).765.2 Points of distinction between the constructionof Kakeya-type sets over Cantor directions inthe plane and over arbitrary sublacunary setsin any dimensionWe have already pointed out some of the added difficulties that arise when trying touse similar arguments to the ones outlined in the previous section in a more generalsetting. We summarize these points here and draw the readers attention to severalothers.The proof of Theorem 1.2 is modeled on the proof of Bateman and Katz’s theorempresented in Section 5.1, with several important distinctions. Overall, our goal inTheorem 1.2 remains essentially the same: to construct a family of tubes rootedon the hyperplane {0} × [0, 1)d, the union of which will eventually give rise to aKakeya-type set. The slopes of the constituent tubes will be assigned from Ω via arandom mechanism involving stickiness akin to the one developed by Bateman andKatz and described in Section 5.1.2. We develop this random mechanism in detailin Chapter 8, with the requisite geometric considerations collected in Chapter 7.As we have already noted, the notion of Bernoulli percolation on trees plays animportant role in the proof of our Theorem 1.2, as it did in the two-dimensionalsetting. The higher-dimensional structure of Ω does however result in minor changesto the argument, as we will see in Chapter 9. But of the two estimates analogousto (5.4) and (5.5) necessary for the Kakeya-type construction, the first is wherethe greatest amount of new work is to be done. The bound (5.4) used in [4] isdeterministic, providing a bound on the size of any sticky collection of tubes asdefined in (5.3). However, the counting argument that led to this bound fails toproduce a tight enough estimate in higher dimensions; instead, we replace it by aprobabilistic statement that suffices for our purposes.More precisely, the issue is the following. A large lower bound on a union oftubes follows if they do not have significant pairwise overlap amongst themselves;i.e. if the total size of pairwise intersections is small. In dimension two, a good upper77bound on the size of this intersection was available uniformly in every sticky slopeassignment. Although the argument that provided this bound is not transferableto general dimensions, it is still possible to obtain the desired bound with largeprobability. A probabilistic statement similar to but not as strong as (5.4) can bederived relatively easily via an estimate on the first moment of the total size ofrandom pairwise intersections. Unfortunately, this is still not sharp enough to yieldthe disparity in the sizes of the tubes and their translated counterparts necessary toclaim the existence of a Kakeya-type set. Indeed, since we prove the straight analogueof the already probabilistic bound (5.5), in order to claim the existence of a singleset satisfying both probabilistic estimates simulatenously, we will need knowledge ofthe variance in sizes of collections of tubes over a not necessarily uniform probabilityspace of sticky maps. Thus, for our higher dimensional setting of Theorem 1.2, weneed a second moment estimate on the pairwise intersections of tubes.Both moment estimates share some common features. For instance, they both ex-ploit Euclidean distance relations between roots and slopes of two intersecting tubes,and combine this knowledge with the relative positions of the roots and slopes withinthe respective trees in which they live, which affects the slope assignments. However,the technicalities are far greater for the second moment compared to the first. Inparticular, for the second moment we are naturally led to consider not just pairs, buttriples and quadruples of tubes, and need to evaluate the probability of obtainingpairwise intersections among these. Not surprisingly, this probability depends on thestructure of the root tuple within its ambient tree. It is the classification of theseroot configurations, computation of the relevant probabilities and their subsequentapplication to the estimation of expected intersections that form the novel pieces ofthe proof of Theorem 1.2 and distinguish it from the planar case.These added complications remain present in the general treatment when weprove Theorem 1.3. However, in this more general setting, we have to adjust ournotion of a sticky mapping between root and slope trees to take better advantageof the potentially very sparse structure of the given sublacunary slope tree. Thiswill lead naturally to a type of mapping between trees that we call weakly sticky ;see Section 13.2. It is this more flexible notion of stickiness between trees that will78allow us to exploit the same general methods developed in the proof of Bateman andKatz’s result presented in this chapter, as well as the methods that we will developduring the proof of Theorem 1.2.79Chapter 6Setup of construction ofKakeya-type sets in Rd+1 over aCantor set of directions: areformulation of Theorem 1.2With this chapter, we begin the program of directly proving our Theorems 1.2 and1.3. Chapter 6 through Chapter 11 cover the construction of Kakeya-type sets overa Cantor set of directions in an arbitrary number of dimensions, Theorem 1.2, whileChapters 12 through 19 treat the general case of Theorem 1.3.To begin, we choose some integer M ≥ 3 and a generalized Cantor-type setCM ⊆ [0, 1) as described in Section 1.1, and fix these items for the remainder. Wealso fix an injective map γ : [0, 1] → {1}×[−1, 1]d satisfying the bi-Lipschitz conditionin (1.4). These objects then define a fixed set of directions Ω = {γ(t) : t ∈ CM} ⊆{1} × [−1, 1]d.Next, we define the collection of tubes that will comprise our Kakeya-type set.LetQ(n) := {t ∈ T ({0} × [0, 1)d;M) : h(t) = n}, (6.1)be the collection of disjoint d-dimensional cubes of sidelength M−n generated by the80lattice M−nZd in the set {0}× [0, 1)d. More specifically, each t ∈ Q(n) is of the formt = {0} ×d∏l=1[jlMn, jl + 1Mn), (6.2)for some j = (j1, . . . , jd) ∈ {0, 1, · · · ,Mn − 1}d, so that #(Q(n)) = Mnd. Fortechnical reasons, we also define Qt to be the cd-dilation of t about its center point,where cd is a small, positive, dimension-dependent constant. The reason for thistechnicality, as well as possible values of cd, will soon emerge in the sequel, but forconcreteness choosing cd = d−2d will suffice.Fix an arbitrarily large integer N ≥ 1, typically much bigger than M . For thesake of establishing Theorem 1.2, we will set n = N in most of what follows, throughChapter 11. We will however prove some more generic facts along the way, forexample in Chapter 7; thus, in these instances we will work with an arbitrary integern. This will allow us to easily apply these facts when we treat the general case ofTheorem 1.3.Recall that the Nth iterate C [N ]M of the Cantor construction is the union of 2Ndisjoint intervals each of length M−N . We choose a representative element of CMfrom each of these intervals, calling the resulting finite collection D[N ]M . Clearlydist(x,D[N ]M ) ≤M−N for every x ∈ CM . SetΩN := γ(D[N ]M ), (6.3)so that dist(ω,ΩN) ≤ CM−N for any ω ∈ Ω, with C as in (1.4). The followingfact is an immediate corollary of Proposition 3.2 that will naturally prove vital inestablishing Theorem 1.2.Fact 6.1. With the set D[N ]M defined as above, TN(D[N ]M ;M) ∼= TN([0, 1); 2).For any t ∈ Q(N) and any ω ∈ ΩN , we definePt,ω := {r + sω : r ∈ Qt, 0 ≤ s ≤ 10C0} , (6.4)81where C0 is a large constant to be determined shortly (for instance, C0 = ddc−1will work, with c as in (1.4)). Thus the set Pt,ω is a cylinder oriented along ω. Its(vertical) cross-section in the plane x1 = 0 is the cube Qt. We say that Pt,ω is rootedat t. While Pt,ω is not strictly speaking a tube as defined in the introduction, thedistinction is negligible, since Pt,ω contains and is contained in constant multiples ofδ-tubes with δ = cd ·M−N . By a slight abuse of terminology but no loss of generality,we will henceforth refer to Pt,ω as a tube.If a slope assignment σ : Q(N) → ΩN has been specified, we set Pt,σ := Pt,σ(t).Thus {Pt,σ : t ∈ Q(N)} is a family of tubes rooted at the elements of an M−N -finegrid in {0} × [0, 1)d, with essentially uniform length in t that is bounded above andbelow by fixed absolute constants. Two such tubes are illustrated in Figure 6.1. Forthe remainder, we setKN(σ) :=⋃t∈Q(N)Pt,σ. (6.5)t1 Pt1,σPt2,σt2Figure 6.1: Two typical tubes Pt1,σ and Pt2,σ rooted respectively at t1 and t2 in the{x1 = 0}–coordinate plane.For a certain choice of sticky slope assignment σ, this collection of tubes will be82shown to generate a Kakeya-type set in the sense of Definition 1.1. This particu-lar slope assignment will not be explicitly described, but rather inferred from thecontents of the following proposition.Proposition 6.2. For any N ≥ 1, let ΣN be a finite collection of slope assignmentsfrom the lattice Q(N) to the direction set ΩN . Every σ ∈ ΣN generates a set KN(σ)as defined in (6.5). Denote the power set of ΣN by P(ΣN).Suppose that (ΣN ,P(ΣN),Pr) is a discrete probability space equipped with theprobability measure Pr, for which the random sets KN(σ) obey the following esti-mates:Pr({σ : |KN(σ) ∩ [0, 1] × Rd| ≥ aN})≥ 34, (6.6)andEσ|KN(σ) ∩ [C0, C0 + 1] × Rd| ≤ bN , (6.7)where C0 ≥ 1 is a fixed constant, and {aN}, {bN} are deterministic sequences satis-fyingaNbN→∞, as N →∞.Then Ω admits Kakeya-type sets.Proof. Fix any integer N ≥ 1. Applying Markov’s Inequality to (6.7), we see thatPr({σ : |KN(σ) ∩ [C0, C0 + 1] × Rd| ≥ 4bN})≤ Eσ|KN(σ) ∩ [C0, C0 + 1] × Rd|4bN≤ 14,so,Pr({σ : |KN(σ) ∩ [C0, C0 + 1] × Rd| ≤ 4bN})≥ 34. (6.8)Combining this estimate with (6.6), we find thatPr({σ : |KN(σ) ∩ [0, 1] × Rd| ≥ aN}⋂{σ : |KN(σ) ∩ [C0, C0 + 1] × Rd| ≤ 4bN})≥ Pr({|KN(σ) ∩ [0, 1] × Rd| ≥ aN})+ Pr({|KN(σ) ∩ [C0, C0 + 1] × Rd| ≤ 4bN})− 1≥ 34+ 34− 1 = 12.83We may therefore choose a particular σ ∈ ΣN for which the size estimates on KN(σ)given by (6.6) and (6.8) hold simultaneously. SetEN := KN(σ) ∩ [C0, C0 + 1] × Rd, so that E∗N(2C0 + 1) ⊇ KN(σ) ∩ [0, 1] × Rd.Then EN is a union of δ-tubes oriented along directions in ΩN ⊂ Ω for which|E∗N(2C0 + 1)||EN |≥ aN4bN→∞, as N →∞,by hypothesis. This shows that Ω admits Kakeya-type sets, per condition (1.1).Proposition 6.2 proves our Theorem 1.2. The following five chapters are devotedto establishing a proper randomization over slope assignments ΣN that will thenallow us to verify the hypotheses of Proposition 6.2 for suitable sequences {aN}and {bN}. We return to a more concrete formulation of the required estimates inProposition 8.4.84Chapter 7Families of intersecting tubesIn this chapter, we will take the opportunity to establish some geometric facts abouttwo intersecting tubes in Euclidean space. These facts will be used in several in-stances within the proof of Theorem 1.2, as well as in our more general Theorem 1.3.Nonetheless they are really general observations that are not limited to our specificarrangement or description of tubes.Lemma 7.1. For v1, v2 ∈ ΩN and t1, t2 ∈ Q(n), t1 6= t2, let Pt1,v1 and Pt2,v2 be thetubes defined as in (6.4). If there exists x = (x1, · · · , xd+1) ∈ Pt1,v1 ∩ Pt2,v2, then theinequality∣∣cen(t2) − cen(t1) + x1(v2 − v1)∣∣ ≤ 2cd√dM−N , (7.1)holds, where cen(t) denotes the centre of the cube t.Proof. The proof is described in the diagram below. If x ∈ Pt1,v1 ∩Pt2,v2 , then thereexist y1 ∈ Qt1 , y2 ∈ Qt2 such that x = y1+x1v1 = y2+x1v2; i.e., x1(v2−v1) = y1−y2.The inequality (7.1) follows since |yi − cen(ti)| ≤ cd√dM−n for i = 1, 2.The inequality in (7.1) provides a valuable tool whenever an intersection takesplace. For the reader who would like to look ahead, Lemma 7.1 will be used alongwith Corollary 7.2 to establish Lemma 10.4. The following Corollary 7.3 will beneeded for the proofs of Lemmas 10.5 and 10.9.85y1Pt1,v1Pt2,v2y2xFigure 7.1: A simple triangle is defined by two rooted tubes, Pt1,v1 and Pt2,v2 , andany point x in their intersection.Corollary 7.2. Under the hypotheses of Lemma 7.1 and for cd > 0 suitably small,|x1(v2 − v1)| ≥12·M−n. (7.2)Proof. Since t1 6= t2, we must have |cen(t1) − cen(t2)| ≥ M−n. Thus an intersectionis possible only ifx1|v2 − v1| ≥ |cen(t2) − cen(t1)| − 2cd√dM−n ≥ (1 − 2cd√d)M−n ≥ 12·M−n,where the first inequality follows from (7.1) and the last inequality holds providedcd is chosen to satisfy 2cd√d ≤ 12 .Corollary 7.3. If t1 ∈ Q(n), v1, v2 ∈ ΩN and a cube Q ⊆ Rd+1 of sidelengthC1M−n with sides parallel to the coordinate axes are given, then there exists at mostC2 = C2(C1) choices of t2 ∈ Q(n) such that Pt1,v1 ∩ Pt2,v2 ∩Q 6= ∅.Proof. As x = (x1, · · · , xd+1) ranges in Q, x1 ranges over an interval I of lengthC1M−n. If x ∈ Pt1,v1 ∩ Pt2,v2 ∩ Q, the inequality (7.1) and the fact diam(Ω) ≤86diam({1} × [−1, 1]d) = 2√d implies∣∣cen(t2) − cen(t1) + cen(I)(v2 − v1)∣∣ ≤ |(x1 − cen(I))(v2 − v1)| + 2cd√dM−n≤ 2√d(C1 + cd)M−n,restricting cen(t2) to lie in a cube of sidelength 2√d(C1+cd)M−n centred at cen(t1)−cen(I)(v2 − v1). Such a cube contains at most C2 subcubes of the form (6.2), andthe result follows.A recurring theme in the proof of Theorem 1.2 is the identification of a criterionthat ensures that a specified point lies in the Kakeya-type set KN(σ) defined in (6.5).With this in mind, we introduce for any x = (x1, x2, · · · , xd+1) ∈ [0, 10C0]×Rd a setPoss(x) :={t ∈ Q(N) : there exists v ∈ ΩN such that x ∈ Pt,v}. (7.3)This set captures all the possible M−N -cubes of the form (6.2) in {0} × [0, 1)d suchthat a tube rooted at one of these cubes has the potential to contain x, providedit is given the correct orientation. Note that Poss(x) is independent of any slopeassignment σ. Depending on the location of x, Poss(x) could be empty. This wouldbe the case if x lies outside a large enough compact subset of [0, 10C0] × Rd, forexample. Even if Poss(x) is not empty, an arbitrary slope assignment σ may notendow any t in Poss(x) with the correct orientation.In the next lemma, we list a few easy properties of Poss(x) that will be helpfullater, particularly during the proof of Lemma 11.3. Lemma 7.4 establishes the mainintuition behind the Poss(x) set, as we give a more geometric description of Poss(x)in terms of an affine copy of the direction set ΩN . This is illustrated in Figure 7.2for a particular choice of directions ΩN .Lemma 7.4. Suppose a slope assignment σ : Q(n) → Ω has been specified.(a) Then we have the containment{t ∈ Q(n) : x ∈ Pt,σ}⊆ Poss(x).87(a)x(b)Figure 7.2: Figure (a) depicts the cone generated by a second stage Cantor construc-tion, Ω2, on the set of directions given by the curve {(1, s, s2) : 0 ≤ s ≤ C} in the{1}×R2 plane. In Figure (b), a point x = (x1, x2, x3) has been fixed and the cone ofdirections has been projected backward from x onto the coordinate plane, x− x1Ω2.The resulting Poss(x) set is thus given by all cubes t ∈ Q(N) such that Qt intersectsa subset of the curve {(0, x2 − x1s, x3 − x1s2) : 0 ≤ s ≤ C}.(b) Further, for any x ∈ [0, 10C0] × Rd,Poss(x) ={t ∈ Q(n) : Qt ∩ (x− x1Ω) 6= ∅}(7.4)⊆ {t ∈ Q(n) : t ∩ (x− x1Ω) 6= ∅}. (7.5)Note that the set in (7.4) could be empty, but the one in (7.5) is not.Proof. If x ∈ Pt,σ, then x ∈ Pt,σ(t) with σ(t) equal to some v ∈ Ω. Thus Pt,v containsx and hence t ∈ Poss(x), proving part (a). For part (b), we observe that x ∈ Pt,v forsome v ∈ Ω if and only if x − x1v ∈ Qt; i.e., Qt ∩ (x − x1Ω) 6= ∅. This proves the88relation (7.4). The containment in (7.5) is obvious.We will also need a bound on the cardinality of Poss(x) within a given cube, andon the cardinality of possible slopes that give rise to indistinguishable tubes passingthrough a given point x sufficiently far away from the root hyperplane. Containedin Lemma 7.5, these results will prove critical throughout Chapter 11, specifically inthe proofs of Lemmas 11.1 and 11.2. We will also have need to formulate a version ofthis lemma in the language of trees: see Lemma 8.3. Not surprisingly, the Cantor-likeconstruction of Ω plays a role in all these estimates.Lemma 7.5. There exists a constant C0 ≥ 1 with the following properties.(a) For any x ∈ [C0, C0 + 1] × Rd and t ∈ Q(N), there exists at most one v ∈ ΩNsuch that p ∈ Pt,v. In other words, for every Qt in Poss(p), there is exactly oneδ-tube rooted at t that contains p.(b) For any p as in (a), and Qt, Qt′ ∈ Poss(p), let v = γ(α), v′ = γ(α′) be the twounique slopes in ΩN guaranteed by (a) such that p ∈ Pt,v∩Pt′,v′. If k is the largestinteger such that Qt and Qt′ are both contained in the same cube Q ⊆ {0}×[0, 1)dof sidelength M−k whose corners lie in M−kZd, then α and α′ belong to the samekth stage basic interval in the Cantor construction.Proof. (a) Suppose v, v′ ∈ ΩN are such that p ∈ Pt,v ∩ Pt,v′ . Then p − p1v andp− p1v′ both lie in Qt, so that p1|v − v′| ≤ cd√dM−N . Since p1 ≥ C0 and (1.4)holds, we find that|α− α′| ≤ cd√dcC0M−N < M−N ,where the last inequality holds if C0 is chosen large enough. Let us recall fromthe description of the Cantor-like construction in Section 1.1 that any two basicrth stage intervals are non-adjacent, and hence any two points in CM lying indistinct basic rth stage intervals are separated by at least M−r. Therefore theinequality above implies that both α and α′ belong to the same basic Nth stageinterval in C[N ]M . But D[N ]M contains exactly one element from each such interval.So α = α′ and hence v = v′.89(b) If p ∈ Pt,v∩Pt′,v′ , then p1|v−v′| ≤ diam( ˜Qt∪ ˜Qt′) ≤ diam(Q) =√dM−k. Apply-ing (1.4) again combined with p1 ≥ C0, we find that |α− α′| ≤√dcC0M−k < M−k,for C0 chosen large enough. By the same property of the Cantor construction asused in (a), we obtain that α and α′ lie in the same kth stage basic interval inC[k]M .We should point out that we will require a direct analogue of Lemma 7.5 whentreating the general case of sublacunary direction sets in Theorem 1.3. We deferthe statement and proof of this analogue until Section 13.1, Lemma 13.1, as it isinstructive to first understand the pruning mechanism that defines our direction setΩN . While a general lemma could easily be stated that encompasses both Lemma 7.5and Lemma 13.1, keeping them separate avoids some unhelpful abstraction.90Chapter 8The random mechanism and stickycollections of tubes in Rd+1 over aCantor set of directionsAs we have seen in the planar context, the construction of a Kakeya-type set withorientations given by Ω will require a certain random mechanism. We now describethis mechanism in detail when Ω arises from a Cantor set of directions in an arbitrarynumber of dimensions.In order to assign a slope σ(·) to the tubes Pt,σ := Pt,σ(t) given by (6.4), we wantto define a collection of random variables {X〈i1,...,ik〉 : 〈i1, . . . , ik〉 ∈ T ([0, 1)d;M)},one on each edge of the tree used to identify the roots of these tubes. The treeT1([0, 1)d) consists of all first generation edges of T ([0, 1)d). It has exactly Md manyedges and we place (independently) a Bernoulli(12) random variable on each edge:X〈0〉, X〈1〉, . . . , X〈Md−1〉. Now, the tree T2([0, 1)d) consists of all first and second gen-eration edges of T ([0, 1)d). It has Md+M2d many edges and we place (independently)a new Bernoulli(12) random variable on each of the M2d second generation edges. Welabel these X〈i1,i2〉 where 0 ≤ i1, i2 < Md. We proceed in this way, eventually assign-ing an ordered collection of independent Bernoulli(12) random variables to the tree91TN([0, 1)d):XN :={X〈i1,...,ik〉 : 〈i1, . . . , ik〉 ∈ TN([0, 1)d), 1 ≤ k ≤ N},where X〈i1,...,ik〉 is assigned to the unique edge identifying 〈i1, i2, · · · , ik〉, namely theedge joining 〈i1, i2, · · · , ik−1〉 to 〈i1, i2, . . . , ik〉. Each realization of XN is a finiteordered collection of cardinality Md +M2d + · · · +MNd with entries either 0 or 1.We will now establish that every realization of the random variable XN defines asticky map between the truncated position tree TN([0, 1)d) and the truncated binarytree TN([0, 1); 2), as defined in Definition 3.1. Fix a particular realization XN = x ={x〈i1,··· ,ik〉}. Define a map τx : TN([0, 1)d) → TN([0, 1); 2), whereτx(〈i1, i2, . . . , ik〉) =〈x〈i1〉, x〈i1,i2〉, . . . , x〈i1,i2,...,ik〉〉. (8.1)We then have the following key proposition.Proposition 8.1. The map τx just defined is sticky for every realization x of XN .Conversely, any sticky map τ between TN([0, 1)d) and TN([0, 1); 2) can be written asτ = τx for some realization x of XN .Proof. Recalling Definition 3.1, we need to verify that τx preserves heights and lin-eages. By (8.1), any finite sequence v = 〈i1, i2, · · · , ik〉 in T ([0, 1)d) is mapped toa sequence of the same length in T ([0, 1); 2). Therefore h(v) = h(τx(v)) for everyv ∈ T ([0, 1)d).Next suppose u ⊃ v. Then u = 〈i1, . . . , ih(u)〉, with h(u) ≤ k. So again by (8.1),τx(u) =〈x〈i1〉, . . . , x〈i1,...,ih(u)〉〉⊃〈x〈i1〉, . . . , x〈i1,...,ih(u)〉, . . . , x〈i1,...,ik〉〉= τx(v).Thus, τx preserves lineages, establishing the first claim in Proposition 8.1.For the second, fix a sticky map τ : TN([0, 1)d) → TN([0, 1); 2). Define x〈i1〉 :=τ(〈i1〉), x〈i1,i2〉 := pi2 ◦ τ(〈i1, i2〉), and in generalx〈i1,··· ,ik〉 := pik ◦ τ(〈i1, i2, · · · , ik〉), k ≥ 1,92where pik denotes the projection map whose image is the kth coordinate of the inputsequence. The collection x = {x〈i1,i2,··· ,ik〉} is the unique realization of XN that verifiesthe second claim.8.1 Slope assignment algorithmRecall from Section 1.1 and Chapter 6 that Ω := γ(CM) and ΩN := γ(D[N ]M ), whereCM is the generalized Cantor-type set and D[N ]M a finitary version of it. In orderto exploit the binary structure of the trees T (CM) := T (CM ;M) and T (D[N ]M ) :=T (D[N ]M ;M) advanced in Proposition 3.2 and Fact 6.1, we need to map traditionalbinary sequences onto the subsequences of {0, . . . ,M − 1}∞ defined by CM or D[N ]M .Proposition 8.2. Every sticky map τ as in (8.1) that maps TN([0, 1)d;M) to TN([0, 1); 2)induces a natural mapping σ = στ from TN([0, 1)d) into ΩN . The maps στ obey auniform Lipschitz-type condition: for any t, t′ ∈ TN([0, 1)d), t 6= t′,∣∣στ (t) − στ (t′)∣∣ ≤ CM−h(D(τ(t),τ(t′))), (8.2)where C is as in (1.4).Remark: While the choice of D[N ]M for a given C[N ]M is not unique, the mapping τ 7→ στis unique given a specific choice. Moreover, if D[N ]M and D[N ]M are two selections offinitary direction sets at scale M−N , then the corresponding maps στ and στ mustobey∣∣στ (v) − στ (v)∣∣ ≤ CM−h(v) for every v ∈ TN([0, 1)d), (8.3)where C is as in (1.4). Thus given τ , the slope in Ω that is assigned by στ to anM -adic cube in {0}× [0, 1)d of sidelength M−N is unique up to an error of O(M−N).As a consequence Pt,στ and Pt,στ are comparable, in the sense that each is containedin a O(M−N)-thickening of the other.Proof. There are two links that allow passage of τ to σ. The first of these is theisomorphism ψ constructed in Proposition 3.2 that maps T (CM ;M) onto T ([0, 1); 2).93Under this isomorphism, the pre-image of any k-long sequence of 0’s and 1’s is avertex w of height k in T (CM ;M), in other words one of the 2k chosen M -adicintervals of length M−k that constitute C[k]M . The second link is a mapping Φ :TN(CM ;M) → D[N ]M that sends every vertex w to a point in CM ∩w, where, as usual,we have also let w denote the particular M -adic interval that it identifies. While thechoice of the image point, i.e., D[N ]M is not unique, any two candidates Φ, Φ satisfy|Φ(w) − Φ(w)∣∣ ≤ diam(w) = M−h(w) for every w ∈ TN(CM ;M). (8.4)We are now ready to describe the assignment τ 7→ σ = στ . Given a sticky mapτ : TN([0, 1)d;M) → TN([0, 1); 2) such thatτ(〈i1, i2, · · · , ik〉) = 〈X〈i1〉, · · · , X〈i1,i2,··· ,ik〉〉,the transformed random variableY〈i1,i2...,ik〉 := γ ◦ Φ ◦ ψ−1 (〈X〈i1〉, X〈i1,i2〉, . . . , X〈i1,i2,...,ik〉〉)associates a random direction in ΩN = γ(D[N ]M ) to the sequence t = 〈i1, . . . , ik〉identified with a unique vertex t ∈ TN([0, 1)d). Thus, definingσ := γ ◦ Φ ◦ ψ−1 ◦ τ (8.5)gives the appropriate (random) mapping claimed by the proposition. The weakLipschitz condition (8.2) is verified as follows,∣∣στ (t) − στ (t′)∣∣ =∣∣γ ◦ Φ ◦ ψ−1 ◦ τ(t) − γ ◦ Φ ◦ ψ−1 ◦ τ(t′)∣∣≤ C∣∣Φ ◦ ψ−1 ◦ τ(t) − Φ ◦ ψ−1 ◦ τ(t′)∣∣≤ CM−h(D(ψ−1◦τ(t),ψ−1◦τ(t′)))= CM−h(D(τ(t),τ(t′))).Here the first inequality follows from (1.4), the second from the definition of Φ.94The third step uses the fact that ψ is an isomorphism, so that h(D(τ(t), τ(t′))) =h(D(ψ−1 ◦τ(t), ψ−1 ◦τ(t′))). Finally, any non-uniqueness in the definition of σ comesfrom Φ, hence (8.3) follows from (8.4) and (1.4).The stickiness of the maps τx is built into their definition (8.1). The reader maybe interested in observing that there is a naturally sticky map that we have alreadyintroduced, which should be viewed as the inspiration for the construction of τ andστ . We refer to the geometric content of Lemma 7.5, which in the language of treeshas a particularly succinct reformulation. We record this below.Lemma 8.3. For C0 obeying the requirement of Lemma 7.5 and x ∈ [C0, C0+1]×Rd,let Poss(x) be as in (7.3). Let Φ and ψ be the maps used in Proposition 8.2. Thenthe map t 7→ β(t) which maps every t ∈ Poss(x) to the unique β(t) ∈ [0, 1) such thatx ∈ Pt,v(t) where v(t) = γ ◦ Φ ◦ ψ−1 ◦ β(t), (8.6)extends as a well-defined sticky map from TN(Poss(x);M) to TN([0, 1); 2).Proof. By Lemma 7.5(a), there exists for every t ∈ Poss(x) a unique v(t) ∈ ΩN suchthat x ∈ Pt,v(t). Let us therefore define for 1 ≤ k ≤ N ,β(pi1(t), · · · , pik(t)) = (pi1 ◦ β(t), · · · , pik ◦ β(t)) (8.7)where β(t) is as in (8.6) and as always pik denotes the projection to the kth coordinateof an input sequence. More precisely, pik(t) represents the unique kth level M -adiccube that contains t. Similarly pik(β(t)) is the kth component of the N -long binarysequence that identifies β(t). The function β defined in (8.7) maps TN(Poss(x);M)to TN([0, 1); 2), and agrees with β as in (8.6) if k = N .To check that the map is consistently defined, we pick t 6= t′ in Poss(x) withu = D(t, t′) and aim to show that β(pi1(t), · · · , pik(t)) = β(pi1(t′), · · · , pik(t′)) for allk such that k ≤ h(u). But by definition (8.6), v(t) and v(t′) have the property thatx ∈ Pt,v(t) ∩ Pt′,v(t′). Hence Lemma 7.5(b) asserts that α(t) = γ−1(v(t)) and α(t′) =γ−1(v(t′)) share the same basic interval at step h(u) of the Cantor construction.95Thus β(t) = ψ ◦ Φ−1 ◦ α(t) and β(t′) = ψ ◦ Φ−1 ◦ α(t′) have a common ancestorin TN([0, 1); 2) at height h(u), and hence pik(β(t)) = pik(β(t′)) for all k ≤ h(u), asclaimed. Preservation of heights and lineages is a consequence of the definition (8.7),and stickiness follows.8.2 Construction of Kakeya-type sets revisitedAs τ ranges over all sticky maps τx : TN([0, 1)d) → TN([0, 1); 2) with x ∈ XN , we nowhave for every vertex t ∈ TN([0, 1)d) with h(t) = N a random sticky slope assignmentσ(t) ∈ ΩN defined as above. For all such t, this generates a randomly oriented tubePt,σ given by (6.4) rooted at the M -adic cube identified by t, with sidelength cd ·M−Nin the {x1 = 0} plane. We may rewrite the collection of such tubes from (6.5) asKN(σ) :=⋃t∈TN ([0,1)d)h(t)=NPt,σ. (8.8)On average, a random collection of tubes with the above described sticky slopeassignment will comprise a Kakeya-type set, as per (1.1). Specifically, we will showin the next chapter that the following proposition holds. In view of Proposition 6.2,this will suffice to prove Theorem 1.2.Proposition 8.4. Suppose (ΣN ,P(ΣN),Pr) is the probability space of sticky mapsdescribed above, equipped with the uniform probability measure. For every σ ∈ ΣN ,there exists a set KN(σ) as defined in (8.8), with tubes oriented in directions fromΩN = γ(D[N ]M ). Then these random sets obey the hypotheses of Proposition 6.2 withaN = cM√logNNand bN =CMN, (8.9)where cM and CM are fixed positive constants depending only on M and d. Thecontent of Proposition 6.2 allows us to conclude that Ω admits Kakeya-type sets.96Chapter 9Slope probabilities and rootconfigurations, Cantor caseHaving established the randomization method for assigning slopes to tubes, we arenow in a position to apply this toward the estimation of probabilities of certain eventsthat will be of interest in the next chapter. Roughly speaking, we wish to computeconditional probabilities that one or more cubes on the root hyperplane are assignedprescribed slopes, provided similar information is available for other cubes.Lemma 9.1. Let us fix v1, v2 ∈ ΩN , so that v1 = γ(α1) and v2 = γ(α2) for uniqueα1, α2 ∈ D[N ]M . We also fix t1, t2 ∈ TN([0, 1)d), h(t1) = h(t2) = N , t1 6= t2. Letus denote by u ∈ TN([0, 1)d) and α ∈ TN(D[N ]M ) the youngest common ancestors of(t1, t2) and (α1, α2) respectively; i.e., u = D(t1, t2), α = D(α1, α2). ThenPr(σ(t2) = v2∣∣σ(t1) = v1)=2−(N−h(u)) if h(u) ≤ h(α),0 otherwise.(9.1)Proof. Keeping in mind the slope assignment as described in (8.5), and the sticki-ness of the map τ as given in Proposition 8.1, the proof can be summarized as inFigure 9.1. Since t1 and t2 must map to v1 = γ(α1) and v2 = γ(α2) under σ = στ ,the sticky map ψ−1 ◦ τ must map t1 and t2 to the Nth stage basic intervals in the97Cantor construction containing α1 and α2 respectively. Since sticky maps preserveheights and lineages, we must have h(α) ≥ h(u). Assuming this, we simply count thenumber of distinct edges on the ray defining t2 that are not common with t1. Themap τ generating σ = στ is defined by a binary choice on every edge in TN([0, 1)d),and the rays given by t1 and t2 agree on their first h(u) edges, so we have exactlyN − h(u) binary choices to make. This is precisely (9.1).D(t1, t2)t2t1D(α1, α2)α2α1Φ ◦ ψ−1 ◦ τFigure 9.1: Diagram of the sticky assignment between the two rays defining t1, t2 ∈TN([0, 1)d) and the two rays defining their assigned slopes α1, α2 ∈ D[N ]M . The boldedges defining t1 are fixed to map to the corresponding bold edges at the same heightdefining α1. This leaves a binary choice to be made at each of the dotted edges alongthe path between D(t1, t2) and t2. We see that t2 is assigned the slope v2 under σif and only if these dotted edges are assigned via Φ ◦ ψ−1 ◦ τ to the dotted edges onthe ray defining α2.More explicitly, if t1 = 〈i1, i2, · · · , iN〉 and t2 = 〈j1, · · · , jN〉, then〈i1, · · · , ih(u)〉 = 〈j1, · · · , jh(u)〉. (9.2)The event of interest may therefore be recast as{σ(t2) = v2∣∣σ(t1) = v1}={τ(j1, · · · , jN) = ψ ◦ Φ−1(α2)∣∣∣τ(i1, · · · , iN) = ψ ◦ Φ−1(α1)}98={〈X〈j1〉, · · · , X〈j1,··· ,jN 〉〉 = ψ ◦ Φ−1(α2)∣∣∣〈X〈i1〉, · · · , X〈i1,··· ,iN 〉〉 = ψ ◦ Φ−1(α1)}={X〈j1,··· ,jk〉 = pik ◦ ψ ◦ Φ−1(α2) for h(u) + 1 ≤ k ≤ N},where pik denotes the kth component of the input sequence. At the second stepabove we have used (8.1) and Proposition 8.2, and the third step uses (9.2). Thelast event then amounts to the agreement of two (N − h(u))-long binary sequences,with an independent, 1/2 chance of agreement at each sequential component. Theprobability of such an event is 2−(N−h(u)), as claimed.The same idea can be iterated to compute more general probabilities. To excludeconfigurations that are not compatible with stickiness, let us agree to call a collection{(t, αt) : t ∈ A, h(t) = h(αt) = N} ⊆ TN([0, 1)d) ×D[N ]M (9.3)of point-slope combinations sticky-admissible if there exists a sticky map τ such thatψ−1 ◦ τ maps t to αt for every t ∈ A. Notice that existence of a sticky τ imposescertain consistency requirements on a sticky-admissible collection (9.3); for exampleh(D(αt, αt′)) ≥ h(D(t, t′)), and more generally h(D(αt : t ∈ A′)) ≥ h(D(A′)) for anyfinite subset A′ ⊆ A.For sticky-admissible configurations, we summarize the main conditional proba-bility of interest below.Lemma 9.2. Let A and B be finite disjoint collections of vertices in TN([0, 1)d) ofheight N . Then for any choice of slopes {vt = γ(αt) : t ∈ A ∪ B} ⊆ ΩN such thatthe collection {(t, αt) : t ∈ A ∪B} is sticky-admissible, the following equation holds:Pr(σ(t) = vt for all t ∈ B∣∣ σ(t) = vt for all t ∈ A)=(12)k(A,B),where k(A,B) is the number of distinct edges in the tree identifying B that are notcommon with the tree identifying A. If {(t, αt) : t ∈ A ∪B} is not sticky-admissible,then the probability is zero.99For the remainder of this chapter, we focus on some special events of the formdealt with in Lemma 9.2 that will be critical to the proof of (6.6). In all these casesof interest #(A),#(B) ≤ 2. As is reasonable to expect, the configuration of the rootcubes within the tree TN([0, 1)d) plays a role in determining k(A,B). While there isa large number of possible configurations, we isolate certain structures that will turnout to be generic enough for our purposes.9.1 Four point root configurationsDefinition 9.3. Let I = {(t1, t2); (t′1, t′2)} be an ordered tuple of four distinct pointsin TN([0, 1)d) of height N such thath(u) ≤ h(u′) where u = D(t1, t2), u′ = D(t′1, t′2). (9.4)We say that I is in type 1 configuration if exactly one of the following conditions issatisfied:(a) either u ∩ u′ = ∅, or(b) u′ ( u, or(c) u = u′ = D(ti, t′j) for all i, j = 1, 2If I satisfying (9.4) is not of type 1, we call it of type 2. An ordered tuple I notsatisfying the inequality in (9.4) is said to be of type j = 1, 2 if I′ = {(t′1, t′2); (t1, t2)}is of the same type.The different structural possibilities are listed in Figure 9.2. The advantageof a type 1 configuration is that, in addition to being overwhelmingly popular, itallows (up to permutations) an easy computation of the quantity k(A,B) describedin Lemma 9.2 if #(A) = #(B) = 2, A ∪ B = {t1, t′1, t2, t′2} and #(A ∩ {t1, t2}) =#(B ∩ {t1, t2}) = 1.100Type 1 Configurationsuu′t1 t2 t′1 t′2(a)uu′t2t1 t′1 t′2(c)uu′t2 t′2 t′1 t1(f)uu′t2 t1 t′1 t′2(d)u = u′t1 t′1 t2 t′2(b)uu′t2 t1 t′1 t′2(e)Figure 9.2: All possible four point configurations of type 1, up to permutations.Lemma 9.4. Let I = {(t1, t2); (t′1, t′2)} obeying (9.4) be in type 1 configuration. Letvi = γ(αi), v′i = γ(α′i), i = 1, 2, be (not necessarily distinct) points in ΩN . Thenthere exist two permutations {i1, i2} and {j1, j2} of {1, 2} such thatPr(σ(ti2) = vi2 , σ(t′j2) = v′j2∣∣σ(ti1) = vi1 , σ(t′j1) = v′j1)=(12)2N−h(u)−h(u′).provided the collection {(ti, αi), (t′i, α′i); i = 1, 2} is sticky-admissible. If the admissi-bility requirement is not met, then the probability is zero.Proof. The proof is best illustrated by referring to the above diagram, Figure 9.2.If u ∩ u′ = ∅, then any two permutations will satisfy the conclusion of the lemma,Figure 9.2(a). In particular, choosing i1 = j1 = 1, i2 = j2 = 2, we see that thenumber of edges in B = {t2, t′2} not shared by A = {t1, t′1} is k(A,B) = (N−h(u))+(N − h(u′)) = 2N − h(u) − h(u′). The same argument applies if u = u′ = D(ti, t′j)for all i, j = 1, 2, Figure 9.2(b).101We turn to the remaining case where u′ ( u. Here there are several possiblitiesfor the relative positions of t1, t2. Suppose first that there is no vertex w on theray joining u and u′ with h(u) < h(w) < h(u′) such that w is an ancestor of t1 ort2. This means that the rays of t1, t2 and u′ follow disjoint paths starting from u,so any choice of permutation suffices, Figure 9.2(c). Suppose next that there is avertex w on the ray joining u and u′ with h(u) < h(w) < h(u′) such that w is anancestor of exactly one of t1, t2, but no descendant of w on this path is an ancestorof either t1 or t2, Figure 9.2(d). In this case, we choose ti1 to be the unique elementof {t1, t2} whose ancestor is w. Note that the ray for ti2 must have split off from uin this case. Any permutation of {t′1, t′2} will then give rise to the desired estimate.If neither of the previous two cases hold, then exactly one of {t1, t2}, say ti1 , is adescendant of u′. If u′ = D(ti1 , t′j) for both j = 1, 2, then again any permutation of{t′1, t′2} works, Figure 9.2(e). Thus the only remaining scenario is where there existsexactly one element in {t′1, t′2}, call it t′j1 , such that h(D(ti1 , t′j1)) > h(u′). In thiscase, we choose A = {ti1 , t′j1} and B = {ti2 , t′j2}, Figure 9.2(f). All cases now resultin k(A,B) = 2N − h(u) − h(u′), completing the proof.Lemma 9.5. Let I = {(t1, t2); (t′1, t′2)} obeying (9.4) be in type 2 configuration. Thenthere exist permutations {i1, i2} and {j1, j2} of {1, 2} for which we have the relationsu1 ⊆ u, u2 ( u with h(u) ≤ h(u1) ≤ h(u2), whereu1 = D(ti1 , t′j1), u2 = D(ti2 , t′j2),and for which the following equality holds:Pr(σ(ti1) = vi1 , σ(t′j1) = v′j1∣∣ σ(ti2) = vi2 , σ(t′j2) = v′j2)=(12)2N−h(u)−h(u1)for any choice of slopes v1, v′1, v2, v′2 ∈ ΩN for which {(ti, αi), (t′i, α′i); i = 1, 2} issticky-admissible.Proof. Since I is of type 2, we know that u = u′, and hence all pairwise youngestcommon ancestors of {t1, t′1, t2, t′2} must lie within u, but that there exist i, j ∈ {1, 2}102Type 2 Configurationsu = u′u1u2t1 t′1 t2 t′2(b)u = u′u1 u2t1 t′1 t2 t′2(a)u = u′u1u2t1 t′1 t2 t′2(c)Figure 9.3: All possible four point configurations of type 2, up to permutations.such that h(D(ti, t′j)) > h(u). Let us set (i2, j2) to be a tuple for which h(D(ti2 , t′j2))is maximal. The height inequalities and containment relations are now obvious, andFigure 9.3 shows that k(A,B) = (N − h(u)) + (N − h(u1)) if A = {ti2 , t′j2} andB = {ti1 , t′j1}.9.2 Three point root configurationsThe arguments in the previous section simplify considerably when there are threeroot cubes instead of four. Since the proofs here are essentially identical to thosepresented in Lemmas 9.4 and 9.5, we simply record the necessary facts with theaccompanying diagram of Figure 9.4.Definition 9.6. Let I = {(t1, t2); (t1, t′2)} be an ordered tuple of three distinct pointsin TN([0, 1)d) of height N such that h(u) ≤ h(u′), where u = D(t1, t2), u′ = D(t1, t′2).We say that I is in type 1 configuration if exactly one of the following two conditionsholds:(a) u′ ( u, or(b) u = u′ = D(t2, t′2).Else I is of type 2, in which case one necessarily has u = u′ and u2 = D(t2, t′2) obeysu2 ( u. If h(u) > h(u′), then the type I is the same as that of I′ = {(t1, t′2); (t1, t2)}.103Type 1 Type 2uu′t1 t′2 t2u = u′t′2t1 t2u = u′u2t1 t2 t′2Figure 9.4: Structural possibilities for three point root configurationsLemma 9.7. Let I = {(t1, t2); (t1, t′2)} be any three-point configuration with h(u) ≤h(u′) in the notation of Definition 9.6, and let v1 = γ(α1), v2 = γ(α2) v′2 = γ(α′2) beslopes in ΩN . ThenPr(σ(t2) = v2, σ(t′2) = v′2∣∣σ(t1) = v1)=(12)2N−h(u)−h(u′) if I is of type 1,(12)2N−h(u)−h(u2) if I is of type 2,provided the point-slope combination {(t1, α1), (t2, α2), (t′2, α′2)} is sticky-admissible.104Chapter 10Proposition 8.4: proof of the lowerbound (6.6)To establish the appropriate lower bound (6.6), we will exploit the general measuretheoretic fact introduced by Bateman and Katz in the planar case (see Lemma 5.1of this document). Recall that this fact quantifies the notion that if a collectionof many thin tubes is to have a large volume, then the intersection of most pairsof tubes should be small. In light of this fact, we will see that the derivation ofinequality (6.6) with the aN specified in (8.9) reduces to the following proposition.Throughout this chapter, all probability statements are understood to take placeon the probability space (ΣN ,P(ΣN),Pr) identified in Proposition 8.4.Proposition 10.1. Fix integers N and R with N  M and N − 110 logM N ≤R ≤ N − 10. Define P ∗t,σ,R to be the portion of Pt,σ contained in the vertical slab[MR−N ,MR+1−N ] × Rd. ThenEσ[∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣]. NM−2N+2R, (10.1)where the implicit constant depends only on M and d.If one can show that with large probability and for all R specified in Proposition10.1, the quantity∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣ is bounded above by the right hand side105of (10.1), then Lemma 5.1 would imply (6.6) with aN =√logN/N . Unfortunately,(10.1) only shows this on average for every R, and hence is too weak a statement topermit such a conclusion. However, with some additional work we are able to upgradethe statement in Proposition 10.1 to a second moment estimate, given below. Whilestill not as strong as the statement mentioned above, this suffices for our purposeswith a smaller choice of aN .Proposition 10.2. Under the same hypotheses as Proposition 10.1, there exists aconstant CM,d > 0 such thatEσ[(∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣)2]≤ C2M,d(NM−2N+2R)2. (10.2)Corollary 10.3. Proposition 10.2 implies (6.6) with aN as in (8.9).Proof. Fix a small constant c1 > 0 such that 2c1 < 110 . By Chebyshev’s inequality,(10.2) implies that there exists a large constant CM,d > 0 such that for every R withc1 logN ≤ N −R ≤ 2c1 logN ,Pr({σ :∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣ ≥ 2CM,dN√logNM−2N+2R})≤Eσ[(∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣)2](2CM,dN√logNM−2N+2R)2≤ 14 logN.Therefore,Pr(2c1 logN⋃N−R=c1 logN{σ :∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣ ≥ CM,dN√logNM−2N+2R})≤ c1 logN4 logN< 14.106In other words, for a class of σ with probability at least 34 ,∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣ ≤ CM,dN√logNM−2N+2Rfor every N − R ∈[c1 logN, 2c1 logN]. For such σ and the chosen range of R, weapply Lemma 5.1 with At = P ∗t,σ,R, m = MNd, for which α = CdMR−NM−Nd, and∑t1,t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣ =[∑t1=t2+∑t1 6=t2]∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣≤ αn+ CM,dN√logNM−2N+2R.MR−N +N√logNM−2N+2R. N√logNM−2N+2R =: L.The last step above uses the specified range of R. Lemma 5.1 now yields that∣∣∣⋃tP ∗t,σ,R∣∣∣&(MR−N)2L∼ 1N√logNfor every N − R ∈[c1 logN, 2c1 logN]. Since {∪tP ∗t,σ,R : R ≥ 0} is a disjointcollection, we obtain∣∣KN(σ) ∩ [0, 1] × Rd∣∣ ≥N−c1 logN∑R=N−2c1 logN∣∣∣⋃tP ∗t,σ,R∣∣∣& logN 1N√logN= aN ,which is the desired conclusion (6.6).10.1 Proof of Proposition 10.1Thus, we are charged with proving Proposition 10.2. We will prove Proposition 10.1first, since it involves many of the same ideas as in the proof of the main proposition,but in a simpler setting. We will need to take advantage of several geometric facts,counting arguments and probability estimates prepared in Chapters 7 and 9 that107will be described shortly. For now, we prescribe the main issues in establishing thebound in (10.1).Proof. Given N and R as in the statement of the proposition, we decompose the slab[MR−N ,MR+1−N ] × Rd into thinner slices Zk, whereZk :=[kMN, k + 1MN]× Rd, MR ≤ k ≤MR+1 − 1.Setting Pt,σ,k := Pt,σ ∩ Zk, we observe that P ∗t,σ,R is an essentially disjoint union of{Pt,σ,k}. Since P ∗t,σ,R is transverse to Zk, we arrive at the estimate∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣ =∑MR≤k<MR+1∑t1 6=t2|Pt1,σ,k ∩ Pt2,σ,k|.M−(d+1)N∑MR≤k<MR+1∑t1 6=t2Tt1t2(k) (10.3).M−(d+1)N∑MR≤k<MR+1∑u∈TN ([0,1)d)h(u)<N∑(t1,t2)∈SuTt1t2(k), (10.4)where Tt1t2(k) is a random variable that equals one if Pt1,σ,k ∩ Pt2,σ,k 6= ∅, and iszero otherwise. At the last step in the above string of inequalities, we have furtherstratified the sum in (t1, t2) in terms of their youngest common ancestor u = D(t1, t2)in the tree TN([0, 1)d), with the index set Su of the innermost sum being defined bySu :={(t1, t2) : t1, t2 ∈ TN([0, 1)d), h(t1) = h(t2) = N, D(t1, t2) = u}.We will prove below in Lemma 10.7 thatEσ[∑(t1,t2)∈SuTt1t2(k)].MR−NM−dh(u)+Nd = MR−dh(u)+N(d−1). (10.5)Plugging this expected count into the last step of (10.4) and simplifying, we108obtainEσ[∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣].∑MR≤k<MR+1−1MR−2N∑u∈TN ([0,1)d)h(u)<NM−dh(u).∑MR≤k<MR+1−1MR−2NN . NM2R−2N ,which is the estimate claimed by Proposition 10.1. At the penultimate step, we haveused the fact that there are Mdr vertices u in TN([0, 1)d) of height r, resulting in∑uM−dh(u) =∑0≤r<NM−drMdr = N. (10.6)10.2 Proof of Proposition 10.2Proof. To establish (10.2), we take a similar route, with some extra care in summingover the (now more numerous) indices. Squaring the expression in (10.3), we obtain[∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣]2≤M−2(d+1)N∑k,k′∈[MR,MR+1)∑t1 6=t2t′1 6=t′2Tt1t2(k)Tt′1t′2(k′)≤ S2 + S3 + S4,where the index i in Si corresponds to the number of distinct points in the tuple{(t1, t2); (t′1, t′2)}. More precisely, for i = 2, 3, 4,Si := M−2(d+1)N∑k,k′∑I∈IiTt1t2(k)Tt′1t′2(k′), where (10.7)Ii :={I = {(t1, t2); (t′1, t′2)}∣∣∣∣∣tj, t′j ∈ TN([0, 1)d), h(tj) = h(t′j) = N ∀j = 1, 2,t1 6= t2, t′1 6= t′2, #({t1, t′1, t2, t′2}) = i}.109The main contribution to the left hand side of (10.2) will be from Eσ(S4), and wewill discuss its estimation in detail. The other terms, whose treatment will be brieflysketched, will turn out to be of smaller size.We decompose I4 = I41 ∪ I42, where I4j is the collection of 4-tuples of distinctpoints {(t1, t2); (t′1, t′2)} that are in configuration of type j = 1, 2, as explained inDefinition 9.3. This results in a corresponding decomposition S4 = S41 + S42. ForS41, we further stratify the sum in terms of u = D(t1, t2) and u′ = D(t′1, t′2), wherewe may assume without loss of generality that h(u) ≤ h(u′). Thus,Eσ(S41)=∑k,k′∑u,u′∈TN ([0,1)d)h(u)≤h(u′)<NEσ(S41(u, u′; k, k′))where (10.8)S41(u, u′; k, k′) := M−2(d+1)N∑I∈I41(u,u′)Tt1t2(k)Tt′1t′2(k′), andI41(u, u′) := {I ∈ I41 : u = D(t1, t2), u′ = D(t′1, t′2)}.In Lemma 10.8 below, we will show thatEσ[S41(u, u′; k, k′)].M−2(d+1)NM2R−d(h(u)+h(u′))+2N(d−1)= M2R−4N−d(h(u)+h(u′)).(10.9)Inserting this back into (10.8), we now follow the same summation steps that led to(10.1) from (10.5). Specifically, applying (10.6) twice, we obtainEσ(S41) .M2R−4N∑k,k′∑u,u′M−d(h(u)+h(u′)).∑k,k′N2M2R−4N . N2M4R−4N ,which is the right hand side of (10.2).Next we turn to S42. Motivated by the configuration type, and after permutationsof {t1, t2} and of {t′1, t′2} if necessary (so that the conclusion of Lemma 9.5 holds),we stratify this sum in terms of u = u′ = D(t1, t2) = D(t′1, t′2), u1 = D(t1, t′1),110u2 = D(t2, t′2), writingS42 =∑k,k′∑u,u1,u2∈TN ([0,1)d)u1,u2⊆uS42(u, u1, u2; k, k′), whereS42(u, u1, u2; k, k′) := M−2(d+1)N∑I∈I42(u,u1,u2)Tt1t2(k)Tt′1t′2(k′), andI42(u, u1, u2) :={I ∈ I42∣∣∣u = D(t1, t2) = D(t′1t′2),u1 = D(t1, t′1), u2 = D(t2, t′2)}(10.10)for given u1, u2 ⊆ u with h(u) ≤ h(u1) ≤ h(u2). For such u, u1, u2, we will prove inLemma 10.9 below thatEσ(S42(u, u1, u2; k, k′)).M−2N−2dh(u2). (10.11)Accepting this estimate for the time being, we complete the estimation of Eσ(S42)as follows,Eσ(S42) .∑k,k′∑u,u1,u2M−2N−2dh(u2).M−2N∑k,k′∑u∑u2⊆uM−2dh(u2)∑u1⊆uh(u1)≤h(u2)1.M−2N∑k,k′∑u∑u2⊆uM−2dh(u2)[Md(h(u2)−h(u))](10.12).M−2N∑k,k′∑uM−dh(u)∑u2⊆uM−dh(u2). NM−2N∑k,k′∑uM−2dh(u) (10.13). NM2R−2N . (10.14)For the range N − R ≤ 12 logM N assured by Proposition 10.2, the last quantityabove is smaller than (NM2R−2N)2. The string of inequalities displayed above involverepeated applications of the fact used to prove (10.6), namely that there are Mdj−dh(u)111cubes of sidelength M−j contained in u. Thus the estimates∑u1⊆uh(u1)≤h(u2)1 .h(u2)∑j=h(u)Md(j−h(u)) .Md(h(u2)−h(u)),∑u2⊆uM−dh(u2) .∑N≥j≥h(u)M−djMd(j−h(u)) . NM−dh(u), and∑uM−2dh(u) =N∑j=0MdjM−2dj =N∑j=0M−dj . 1were used in (10.12) (10.13) and (10.14) respectively, completing the estimation ofE(S4).Arguments similar to and in fact simpler than those above lead to the followingestimates for E(S3) and E(S2), where S3 and S2 are as defined in (10.7):E(S3) = E(S31) + E(S32). NM3R−3N +M3R−3N . NM3R−3N , and (10.15)E(S2) . NM3R−(d+3)N . (10.16)Here without loss of generality and after a permutation if necessary, we have assumedthat I = {(t1, t2); (t1, t′2)} ∈ I3, with h(D(t1, t2)) ≤ h(D(t1, t′2)). The subsum S3ithen corresponds to tuples I that are in type i configuration in the sense of Definition9.6. There is only one possible configuration of pairs in I2. The derivation of theexpectation estimates (10.15) and (10.16) closely follow the estimation of S4, withappropriate adjustments in the probability counts; for instance, (10.15) uses Lemma9.7 and (10.16) uses Lemma 9.1. To avoid repetition, we leave the details of (10.15)and (10.16) to the reader, noting that the right hand term in each case is dominatedby (NM2R−2N)2 by our conditions on R.11210.3 Expected intersection countsIt remains to establish (10.5), (10.9) and (10.11). The necessary steps for this arelaid out in the following sequence of lemmas. Unless otherwise stated, we will beusing the notation introduced in the proofs of Propositions 10.1 and 10.2.Lemma 10.4. Fix Zk. Let us define Au = Au(k) to be the (deterministic) collectionof all t1 ∈ TN([0, 1)d), h(t1) = N that are contained in the cube u and whose distancefrom the boundary of some child of u is . kM−N−h(u).For t1 ∈ Au, let Bt1 = Bt1(k) denote the (also deterministic) collection of t2 ∈TN([0, 1)d) with h(t2) = N and D(t1, t2) = u such that the distance between thecentres of t1 and t2 is . kM−N−h(u).(a) Then for any slope assignment σ, the random variable Tt1t2(k) = 0 unless t1 ∈ Auand t2 ∈ Bt1. In other words,∑(t1,t2)∈SuTt1t2(k) =∑t1∈Au∑t2∈Bt1Tt1t2(k), so thatEσ[∑(t1,t2)∈SuTt1t2(k)]=∑t1∈AuEσ[∑t2∈Bt1Tt1t2(k)]. (10.17)(b) The description of Au yields the following bound on its cardinality:#(Au) .( kMN)Md(N−h(u)) .MR−dh(u)+(d−1)N .Proof. We observe that Tt1t2(k) = 1 if and only if there exists a point p = (p1, · · · , pd+1) ∈Zk and v1, v2 ∈ ΩN such that p ∈ Pt1,v1∩Pt2,v2 , and σ(t1) = v1, σ(t2) = v2. By Lemma7.1, this implies that|cen(t1) − cen(t2) + p1(σ(t1) − σ(t2))| ≤ 2cd√dM−N , (10.18)113M−h(u)kM−N−h(u)Figure 10.1: A diagram of Au when d = 2, M = 3. Here the largest square is u.The thatched area depicts Au. The finest squares are the root cubes contained inAu.where cen(ti) denotes the centre of the cube ti. For p ∈ Zk, (10.18) yields|cen(t1) − cen(t2)| ≤ p1|σ(t1) − σ(t2)| + 2cd√dM−N . p1|σ(t1) − σ(t2)|.(k + 1MN)|σ(t1) − σ(t2)| .( kMN)M−h(D(τ(t1),τ(t2))). kM−N−h(u). (10.19)The second inequality in the steps above follows from Corollary 7.2, the third fromthe definition of Zk and the fourth from the property (8.2) of the slope assignment.Here τ is the unique sticky map that generates σ, as specified in Proposition 8.2.Since τ preserves heights and lineages, h(D(τ(t1), τ(t2))) ≥ h(D(t1, t2)) = h(u), andthe last step follows.The inequality in (10.19) implies that Tt1t2(k) = 0 unless t2 ∈ Bt1 . Further, t1, t2114lie in distinct children of u, so t1 must satisfydist(t1, ∂u′) .kMNM−h(u) for some child u′ of u,to allow for the existence of some t2 obeying (10.19). This means t1 ∈ Au, proving(a).For (b) we observe that u has Md children. The Lebesgue measure of the set⋃u′{x ∈ u′ : dist(x, ∂u′) . kM−N−h(u), u′ is a child of u}(10.20)is therefore . (Md)kM−N−h(u)M−(d−1)h(u). The cardinality of Au is comparable tothe number of M−N -separated points in the set (10.20), and (b) follows.Our next task is to make further reductions to the expression on the right handside of (10.17) that will enable us to invoke the probability estimates from Chapter9. To this end, let us fix Zk, t1 ∈ Au(k), v1 = γ(α1) ∈ ΩN , and define a collection ofpoint-slope pairsEu(t1, v1; k) :=(t2, v2)∣∣∣∣∣t2 ∈ TN([0, 1)d) ∩ Bt1 , v2 = γ(α2) ∈ ΩN ,h(t2) = h(α2) = N, u = D(t1, t2),Pt1,v1 ∩ Pt2,v2 ∩ Zk 6= ∅, h(D(α1, α2)) ≥ h(u). (10.21)Thus Eu(t1, v1; k) is non-random as well. The significance of this collection is clarifiedin the next lemma.Lemma 10.5. For (t2, v2) ∈ Eu(t1, v1; k) described as in (10.21), define a randomvariable T t2v2(t1, v1; k) as follows:T t2v2(t1, v1; k) :=1 if σ(t2) = v2,0 otherwise.(10.22)115(a) The random variables Tt1t2(k) and T t2v2(t1, v1; k) are related as follows: givenσ(t1) = v1,Tt1t2(k) = sup{T t2v2(t1, v1; k) : (t2, v2) ∈ Eu(t1, v1; k)}. (10.23)In particular under the same conditional hypothesis σ(t1) = v1, one obtains theboundTt1t2(k) ≤∑v2∈ΩN(t2,v2)∈Eu(t1,v1;k)T t2v2(t1, v1; k), (10.24)which in turn impliesEσ[∑t2∈Bt1Tt1t2(k)∣∣∣σ(t1) = v1]≤∑(t2,v2)∈Eu(t1,v1;k)Pr(σ(t2) = v2∣∣σ(t1) = v1).(10.25)(b) The cardinality of Eu(t1, v1; k) is . 2N−h(u).Proof. We already know from Lemma 10.4 that Tt1t2(k) = 0 unless t2 ∈ Bt1 . Further,if σ(t1) = v1 is known, then it is clear that Tt1t2(k) = 1 if and only if there existsv2 ∈ ΩN such that Pt1,v1 ∩ Pt2,v2 ∩ Zk 6= ∅ and σ(t2) = v2. But this means that thesticky map τ that generates σ must map t2 to the N -long binary sequence that iden-tifies α2. Stickiness dictates that h(D(α1, α2)) = h(D(τ(t1), τ(t2))) ≥ h(D(t1, t2)) =h(u), explaining the constraints that define Eu(t1, v1; k). Rephrasing the discussionabove, given σ(t1) = v1, the event Tt1t2(k) = 1 holds if and only if there existsv2 ∈ ΩN such that (t2, v2) ∈ Eu(t1, v1; k) and σ(t2) = v2. This is the identity claimedin (10.23) of part (a). The bound in (10.24) follows easily from (10.23) since thesupremum is dominated by the sum. The final estimate (10.25) in part (a) fol-lows by taking conditional expectation of both sides of (10.24), and observing thatEσ(T t2v2(t1, v1; k)|σ(t1) = v1) = Pr(σ(t2) = v2∣∣σ(t1) = v1).We turn to (b). If v2 ∈ ΩN is fixed, then it follows from Corollary 7.3 (taking Qin that corollary to be the cube of sidelength O(M−N) containing Pt1,v1 ∩ Zk) thatthere exist at most a constant number of choices of t2 such that (t2, v2) ∈ Eu(t1, v1; k).116But by Fact 6.1 the number of points α2 ∈ D[N ]M (and hence slopes v2 ∈ ΩN) thatobey h(D(α1, α2)) ≥ h(u) is no more than 2N−h(u), proving the claim.The same argument above applied twice yields the following conclusion, the ver-ification of which is left to the reader.Corollary 10.6. Given t1 ∈ Au(k), t′1 ∈ Au′(k′), v1, v′1 ∈ ΩN , define Eu(t1, v1; k) andEu′(t′1, v′1; k′) as in (10.21) and the random variables T t2v2(t1, v1; k), T t′2v′2(t′1, v′1; k′)as in (10.22). Then given σ(t1) = v1 and σ(t′1) = v′1,∑t2∈Bt1t′2∈Bt′1Tt1t2(k)Tt′1t′2(k′) ≤∗∑T t2v2(t1, v1; k)T t′2v′2(t′1, v′1; k′),where the notation∗∑represents the sum over all indices {(t2, v2); (t′2, v′2)} ∈ Eu(t1, v1; k)×Eu′(t′1, v′1; k′).We are now ready to establish the key estimates in the proofs of Propositions10.1 and 10.2.Lemma 10.7. The estimate in (10.5) holds.Proof. We combine the steps outlined in Lemmas 10.4, 10.5 and 9.1. By Lemma10.4(a),Eσ[∑(t1,t2)∈SuTt1t2(k)]=∑t1∈AuEσ[∑t2∈Bt1Tt1t2(k)]=∑t1∈AuEv1Eσ[∑t2∈Bt1Tt1t2(k)∣∣∣σ(t1) = v1].(10.26)Applying (10.25) from Lemma 10.5 followed by Lemma 9.1, we find that the innerexpectation above obeys the boundEσ[∑t2∈Bt1Tt1t2(k)∣∣σ(t1) = v1]≤∑(t2,v2)∈Eu(t1,v1;k)Pr(σ(t2) = v2|σ(t1) = v1)117≤ #(Eu(t1, v1; k)) × 2−N+h(u)︸ ︷︷ ︸Lemma 9.1. 2N−h(u)︸ ︷︷ ︸Lemma 10.5(b)×2−N+h(u) . 1,uniformly in v1. Inserting this back into (10.26), we arrive atEσ[∑(t1,t2)∈SuTt1t2(k)]. #(Au),which according to Lemma 10.4(b) is the bound claimed in (10.5).Lemma 10.8. The estimate in (10.9) holds.Proof. The proof of (10.9) shares many similarities with that of Lemma 10.7, exceptthat there are now two copies of each of the objects appearing in the proof of (10.5)and the probability estimate comes from Lemma 9.4 instead of Lemma 9.1. Weoutline the main steps below.In view of Lemma 9.4 and after a permutation of (t1, t2) and of (t′1, t′2) if necessary,we may assume that for every I = {(t1, t2); (t′1, t′2)} ∈ I41(u, u′),Pr(σ(t2) = v2, σ(t′2) = v′2|σ(t1) = v1, σ(t′1) = v′1)=(12)2N−h(u)−h(u′). (10.27)Now,Eσ(S41(u, u′; k, k′))≤M−2(d+1)NEσ[∑I∈I41(u,u′)Tt1t2(k)Tt′1t′2(k′)]= M−2(d+1)N∑t1∈Au(k)t′1∈Au′ (k′)Ev1,v′1Eσ[∑t2∈Bt1t′2∈Bt′1Tt1t2(k)Tt′1t′2(k′)∣∣∣σ(t1) = v1, σ(t′1) = v′1].M−2(d+1)N(kk′M2NMd(2N−h(u)−h(u′)))︸ ︷︷ ︸#(t1,t′1) from Lemma 10.4.M2R−4N−d(h(u)+h(u′)),118since according to Corollary 10.6Eσ[∑(t2,t′2)∈Bt1×Bt′1Tt1t2(k)Tt′1t′2(k′)∣∣∣σ(t1) = v1, σ(t′1) = v′1]≤ Eσ[∗∑T t2v2(t1, v1; k)T t′2,v′2(t′1, v′1; k′)∣∣∣σ(t1) = v1, σ(t′1) = v′1].∗∑Pr(σ(t2) = v2, σ(t′2) = v′2 | σ(t1) = v1, σ(t′1) = v′1). (2N−h(u))︸ ︷︷ ︸#(Eu(t1,v1;k))× (2N−h(u′))︸ ︷︷ ︸#(Eu′ (t′1,v′1;k′))× (2−2N+h(u)+h(u′))︸ ︷︷ ︸(10.27) via Lemma 9.4. 1, uniformly in v1, v′1.The proof is therefore complete.Lemma 10.9. The estimate in (10.11) holds.Proof. The proof of (10.11) is similar to (10.9), and in certain respects simpler. Butthe configuration type dictates that we set up a different class E∗ of point-slope tuplesthat will play a role analogous to Eu(t1, v1; k) in the preceding lemmas. Recall thestructure of a type 2 configuration from Figure 9.3 and the definition of I42(u, u1, u2)from (10.10). Given root cubes t2, t′2, and u, u1, u2 ∈ TN([0, 1)d) with the propertythatu1 ⊆ u, u2 ( u, u2 = D(t2, t′2), h(u) ≤ h(u1) ≤ h(u2) ≤ N = h(t2) = h(t′2),and slopes v2 = γ(α2), v′2 = γ(α′2) ∈ ΩN , we define E∗ (depending on all theseobjects) to be the following collection of root-slope tuples:E∗ :={(t1, v1); (t′1, v′1)}∣∣∣∣∣I = {(t1, t2); (t′1, t′2)} ∈ I42(u, u1, u2),v1 = γ(α1), v′1 = γ(α′1) for some α1, α′1 ∈ D[N ]M ,Pt1,v1 ∩ Pt2,v2 ∩ Zk 6= ∅, Pt′1,v′1 ∩ Pt′2,v′2 ∩ Zk′ 6= ∅,{(ti, αi), (t′i, α′i) : i = 1, 2} is sticky-admissible .(10.28)119The relevance of E∗ is this: if σ(t2) = v2 and σ(t′2) = v′2 are given, then Tt1t2(k)Tt′1t′2(k′) =0 unless there exist v1, v′1 ∈ ΩN with {(t1, v1); (t′1, v′1)} ∈ E∗ and σ(t1) = v1, σ(t′1) =v′1.We first set about obtaining a bound on the size of E∗ that we will need mo-mentarily. Stickiness dictates that h(D(α1, α2)) ≥ h(u), and that α1 is an Nth leveldescendant of α, the ancestor of α2 at height h(u). Thus the number of possible α1(and hence v1) is ≤ 2N−h(u), by Fact 6.1. Again by stickiness, h(D(α1, α′1)) ≥ h(u1),so for a given α1, the number of α′1 (hence v′1) is no more than the number of possibledescendants of α∗, the ancestor of α1 at height h(u1). This number is thus ≤ 2N−h(u1).Once v1, v′1 have been fixed (recall that v2, v′2, t2, t′2 are already fixed), it follows fromCorollary 7.3 that the number of t1, t′1 obeying the intersection conditions in (10.28)is . 1. Combining these, we arrive at the following bound on the cardinality of E∗:#(E∗) .(2N−h(u))(2N−h(u1))= 22N−h(u)−h(u1). (10.29)We use this bound on the size of E∗ to estimate a conditional expectation, essen-tially the same way as in the previous two lemmas.Eσ[∑t1,t′1I∈I42(u,u1,u2)Tt1t2(k)Tt′1t′2(k′)∣∣σ(t2) = v2, σ(t′2) = v′2]=∑E∗Pr(σ(t1) = v1, σ(t′1) = v′1|σ(t2) = v2, σ(t′2) = v′2). #(E∗)(12)2N−h(u)−h(u1). 1, (10.30)where the last step follows by combining Lemma 9.5 with (10.29). As a result, weobtainEσ(S42(u, u1, u2; k, k′))= M−2(d+1)NEσ[∑I∈I42(u,u1,u2)Tt1t2(k)Tt′1t′2(k′)]120≤M−2(d+1)N∑t2,t′2⊆u2Ev2,v′2Eσ[∑t1,t′1I∈I42(u,u1,u2)Tt1t2(k)Tt′1t′2(k′)∣∣σ(t2) = v2, σ(t′2) = v′2].M−2(d+1)N∑t2,t′2⊆u21.M−2(d+1)N(M−dh(u2)+Nd)2,where the estimate from (10.30) has been inserted in the third step above. The finalexpression is the bound claimed in (10.11).121Chapter 11Proposition 8.4: proof of theupper bound (6.7)Using the theory developed in Chapter 4, we can establish inequality (6.7) withbN = CM/N as in Proposition 8.4 with relative ease. For x ∈ Rd+1, we writex = (x1, x), where x = (x2, . . . , xd+1). Since the Kakeya-type set defined by (8.8) iscontained in the parallelepiped [C0, C0 + 1] × [−2C0, 2C0]d , we may writeEσ∣∣KN(σ) ∩ [C0, C0 + 1] × Rd∣∣ = Eσ(∫ C0+1C0∫[−2C0,2C0]d1KN (σ)(x1, x)dxdx1)=∫ C0+1C0∫[−2C0,2C0]dEσ(1KN (σ)(x1, x))dxdx1=∫ C0+1C0∫[−2C0,2C0]dPr(x) dxdx1, (11.1)where Pr(x) denotes the probability that the point (x1, x) is contained in the setKN(σ). To establish inequality (6.7) then, it suffices to show that this probability isbounded by a constant multiple of 1/N , the constant being uniform in x ∈ [C0, C0 +1] × Rd.Let us recall the definition of Poss(x) from (7.3). We would like to define acertain percolation process on the tree TN(Poss(x)) whose probability of survival can122majorize Pr(x). By Lemma 7.5(a), there corresponds to every t ∈ Poss(x) exactlyone v(t) ∈ ΩN such that Pt,v(t) contains x. Let us also recall that v(t) = γ(α(t))for some α(t) ∈ D[N ]M . By Fact 6.1, α(t) is uniquely identified by β(t) := ψ(α(t)),which is a deterministic sequence of length N with entries 0 or 1. Here ψ is the treeisomorphism described in Lemma 3.2.Given a slope assignment σ = στ generated by a sticky map τ : TN([0, 1)d) →TN([0, 1); 2) as defined in Proposition 8.2 and a vertex t = 〈i1, · · · , iN〉 ∈ TN(Poss(x))with h(t) = N , we assign a value of 0 or 1 to each edge of the ray identifying t asfollows. Let e be the edge identified by the vertex 〈i1, i2, · · · , ik〉. SetYe :={1 if pik(τ(t)) = pik(β(t)),0 if pik(τ(t)) 6= pik(β(t)).(11.2)To clarify the notation above, recall that both τ(t) and β(t) are N -long binarysequences, and pik denotes the kth component of the input. Though the definitionof Ye suggests a potential conflict for different choices of t, our next lemma confirmsthat this is not the case.Lemma 11.1. The description in (11.2) is consistent in t; i.e., it assigns a uniquelydefined binary random variable Ye to each edge of TN(Poss(x)). The collection {Ye}is independent and identically distributed as Bernoulli(12) random variables.Proof. Let t, t′ ∈ TN(Poss(x)), h(t) = h(t′) = N . Set u = D(t, t′), the youngestcommon ancestor of t and t′. In order to verify consistency, we need to ascertainthat for every edge e in TN(Poss(x)) leading up to u and for every sticky map τ , theprescription (11.2) yields the same value of Ye whether we use t or t′. Rephrasingthis, it suffices to establish thatpik(τ(t)) = pik(τ(t′)) and pik(β(t)) = pik(β(t′)) for all 0 ≤ k ≤ h(u). (11.3)Both equalities are consequences of the height and lineage-preserving property of123sticky maps, by virtue of whichh(D(t, t′)) ≤ min[h(D(τ(t), τ(t′))), h(D(β(t), β(t′)))].Of these, stickiness of τ has been proven in Proposition 8.1. The unambiguousdefinition and stickiness of β has been verified in Lemma 8.3.For the remainder, we recall from Chapter 8 (see the discussion preceding Propo-sition 8.1) that for t = 〈i1, i2, · · · , iN〉, the projection pik(τ(t)) = X〈i1,··· ,ik〉 is aBernoulli(12) random variable, so Pr(Ye = 1) =12 . Further the random variables Yeassociated with distinct edges e in TN(Poss(x)) are determined by distinct Bernoullirandom variables of the form X〈i1,··· ,ik〉. The stated independence of the latter col-lection implies the same for the former.Thus the collection YN = {Ye}e∈E defines a Bernoulli percolation on TN(Poss(x)),where E is the edge set of TN(Poss(x)). As described in Section 4.1, the event{Ye = 0} corresponds to the removal of the edge e from E , and the event {Ye = 1}corresponds to retaining this edge.Lemma 11.2. Let Pr(x) = Pr{τ : x ∈ KN(στ )} be as in (11.1), and {Ye} as in(11.2).(a) For any x ∈ [C0, C0 + 1] × Rd, the event {τ : x ∈ KN(στ )} is contained in{τ : ∃ a full-length ray in TN(Poss(x)) that survives percolation via {Ye}}.(11.4)(b) As a result,Pr(x) ≤ Pr(survival after percolation on TN(Poss(x))).Proof. It is clear that x ∈ KN(στ ) if and only if there exists t ∈ Poss(x) such thatστ (t) = v(t), where v(t) is the unique slope in ΩN prescribed by Lemma 7.5(a) for124which x ∈ Pt,v(t). In other words, we have{τ : x ∈ KN(στ )} =⋃{σ(t) = v(t) : t ∈ Poss(x)}=⋃{τ(t) = β(t) : t ∈ Poss(x)}, (11.5)where the last step follows from the preceding one by unraveling the string of bijectivemappings γ−1, Φ−1 and ψ (described in Proposition 8.2) that leads from σ(t) to τ(t),and which incidentally also generates β(t) = 〈j1, · · · , jN〉 ∈ T ([0, 1); 2) from v(t).Since t is identified by some sequence 〈i1, i2, . . . , iN〉, we have its associated randombinary sequenceτ(t) = 〈X〈i1〉, X〈i1,i2〉, . . . , X〈i1,i2,...,iN 〉〉 ∈ TN([0, 1); 2).Using this, we can rewrite (11.5) as follows:⋃t∈Poss(x){σ(t) = v(t)}=⋃t∈Poss(x){〈X〈i1〉, X〈i1,i2〉, . . . , X〈i1,i2,...,iN 〉〉 = 〈j1, j2, . . . , jN〉}=⋃t∈Poss(x)N⋂k=1{X〈i1,...,ik〉 = jk}=⋃R↔〈i1,··· ,iN 〉∈∂T⋂e↔〈i1,...,ik〉∈E∩R{X〈i1,...,ik〉 − jk = 0}=⋃R∈∂T⋂e∈E∩R{Ye = 1}. (11.6)In the above steps we have set T := TN(Poss(x)) for brevity and let E be the edgeset of T . The last step uses (11.2), and the final event is the same as the one in(11.4). Using (11.6), we havePr(x) ≤ Pr(⋃R∈∂T⋂e∈E∩R{Ye = 1}). (11.7)125This last expression is obviously equivalent to the right hand side of (4.1), verifyingthe second part of the lemma.Our next task is therefore to estimate the survival probability of TN(Poss(x))under Bernoulli(12) percolation. For this purpose and in view of the discussion inSection 4.3, we should visualize TN(Poss(x)) as an electrical circuit, the resistance ofan edge terminating at a vertex of height k being 2k−1, per equation (4.2). Let usdenote by R(Poss(x)) the resistance of the entire circuit. In light of the theorem ofLyons, restated in the form of Proposition 4.4, it suffices to establish the followinglemma.Lemma 11.3. With the resistance of Poss(x) defined as above, we haveR(Poss(x)) & N. (11.8)Proof. We follow the same argument as Bateman and Katz [4]. Recalling the con-tainment (7.5) from Lemma 7.4, we find that Nk, the number of kth generationvertices of TN(Poss(x)), is bounded above by Nk, the number of kth level verticesin TN({0} × [0, 1)d ∩ (x − x1ΩN)). We will shortly prove in Lemma 11.4(b) thatNk . 2k, where the implicit constant is uniform in x ∈ [C0, C0 + 1] × [−2C0, 2C0]d.Thus, by Corollary 4.2,R(Poss(x)) ≥N∑k=12k−1Nk&N∑k=12kNK& N,establishing inequality (11.8).The remaining Lemma 11.4 is an easy observation that follows simply from theconnection between sets and the trees used to encode them.Lemma 11.4. Let ΩN be the set defined in (6.3).(a) Given ΩN , there is a constant C1 > 0 (depending only on d and C, c from (1.4))such that for any 1 ≤ k ≤ N , the number of kth generation vertices in TN(ΩN ;M)is ≤ C12k.126(b) For any compact set K ⊆ Rd+1, there exists a constant C(K) > 0 with thefollowing property. For any x = (x1, · · · , xd+1) ∈ K, and 1 ≤ k ≤ N , thenumber of kth generation vertices in TN(E(x);M) is ≤ C(K)2k, where E(x) :=(x− x1ΩN) ∩ {0} × [0, 1)d.Proof. There are exactly 2k basic intervals of level k that comprise C[k]M . Under γ,each such basic interval maps into a set of diameter at most CM−k. Since ΩN =γ(D[N ]M ) ⊆ γ(C[k]M ), the number of kth generation vertices in TN(ΩN ;M), which is alsothe number of kth level M -adic cubes needed to cover ΩN , is at most C12k. Thisproves (a).Let Q be any kth generation M -adic cube such that Q ∩ ΩN 6= ∅. Then on onehand, (x−x1Q)∩(x−x1ΩN) 6= ∅; on the other hand, the number of kth level M -adiccubes covering (x− x1Q) is ≤ C(K), and part (b) follows.Combining Lemmas 11.2 and 11.3 with Proposition 4.4 gives us the desired boundof . 1/N on (11.1). This completes the proof of inequality (6.7), and so too Propo-sition 8.4. Our first Theorem 1.2 is therefore established.127Chapter 12Construction of Kakeya-type setsin Rd+1 over an arbitrarysublacunary set of directionsWith Theorem 1.2 proven, we turn our attention to the more general Theorem 1.3.Recall that this means we will be occupied with an arbitrary set of sublacunary direc-tions, of which a Cantor-type set of directions is a particular example, as describedin Definition 2.7.The method of proof will be analogous to that of the Cantor case, although manyof the details will require a necessarily more technical treatment. Most notably, itwill require substantially more work to define a coherent and flexible enough randommechanism on the slope assignments for a collection of root cubes. We will naturallywant to exploit the geometric idea of stickiness, but to get the full strength of ourTheorem 1.3, we will have to weaken the notion of tree stickiness that was so integralto the proof of Theorem 1.2.It will be convenient to work with a pruned subset of a sublacunary direction set,one that is guaranteed to have certain useful structural properties that we can thenexploit in our subsequent calculations.12812.1 Pruning of the slope treeRecall that to establish Theorem 1.3 it suffices to prove Proposition 3.7; we focusour attention thusly. Fix a base integer M ≥ 2 and a sublacunary direction setΩ ⊆ Rd+1 (obeying the conclusion of Proposition 3.6). We also fix an absoluteconstant C0 ≥ 1, which will remain unchanged for the rest of the proof, and whosevalue will be specified later (C0 = 10 will do). Given any integer N , however large,Proposition 3.6 (see (3.11)) supplies a hyperplane VN at unit distance from the origin,a coordinate system ϕN on VN , and a relatively compact subset WN ⊆ CΩ ∩ VN forwhich split(T (ϕN(WN);M)) > (N + 1)(2C0 + 1)d. The choice of N , and hence VN ,WN and ϕN will stay fixed during the analysis in Chapters 13–19. The existence ofKakeya-type sets, which is the goal of Proposition 3.7, relies on the ability to conductthis analysis for arbitrarily large N . The constant C0, on the other hand, does notchange with N .Without loss of generality we will assume that VN = {1}×Rd and that ϕN is theambient coordinate system in VN (and hence in all hyperplanes parallel to VN). Theuse of ϕN will be dropped in the sequel, and we will simply write split(T (WN ;M)) >(N + 1)(2C0 + 1)d. We will also assume that WN ⊆ {1} × [0, 1)d; indeed if WN ⊆{1} × [0,ML)d for some large L, then we scale by a factor of M−L in directionsperpendicular to e1 = (1, 0, · · · , 0), leaving the direction e1 unchanged. The treecorresponding to the scaled version of WN has the same splitting number as theoriginal tree. Further, a union EN of tubes pointing in the scaled directions canbe rescaled back to tubes with orientations in WN , with the ratio |E∗N |/|EN | (asexplained in (1.1)) unchanged. From this point onwards, our direction set will be anappropriately chosen subset of WN ⊆ {1} × [0, 1)d for a fixed N . We rename WN asΩ, since this will not cause any confusion in the sequel.We now prune our direction set Ω down to a representative tree enjoying specialstructural properties, in terms of M -adic and Euclidean distances between certainvertices. The essential features of this trimming process and the modified directionset are summarized below in the main result of this section.Proposition 12.1. Let M ≥ 2 be a base integer, C0 ≥ 1 a fixed constant, and N  1129a large parameter as described above. Let Ω ⊆ {1}× [0, 1)d be a direction set obeyingthe hypothesis split(T (Ω;M)) > (N + 1)(2C0 + 1)d. Then there exist• a finite subset ΩN ⊆ Ω of cardinality 2N , and• an integer J = J(Ω, N) ≥ Nsuch that the following properties hold for the tree TJ(ΩN ;M) of height J encodingΩN :(i) Every ray in TJ(ΩN ;M) splits exactly N times.(ii) Every splitting vertex in TJ(ΩN ;M) has exactly two children.(iii) For any splitting vertex v of TJ(ΩN ;M), there exists an integer h∗v > h(v)obeying the following constraints:- None of the descendants of the two children of v specified in part (ii) splitat any height strictly smaller than h∗v.- If w1(v), w2(v) are the two descendants of v at height h∗v, then the Eu-clidean distance between the cubes w1 and w2 obeys the relationC0M−h∗v ≤ dist(w1(v), w2(v)) ≤ (2C0 +√d)M−h∗v+1. (12.1)In fact, h∗v is the smallest integer exceeding h(v) with this property.The integer J can be chosen to ensure that the following additional condition is met:(iv) C0M−J ≤ min{|ω − ω′| : ω 6= ω′, ω, ω′ ∈ ΩN}.Notice that the tree TN(D[N ]M ;M) encoding a Cantor-type set of directions that weconsidered in Chapters 6 to 11 already satisfies the requirements of Proposition 12.1by virtue of Fact 6.1; i.e. the slope tree TN(D[N ]M ;M) is already pruned. In this way,we see in what sense a pruned tree behaves like a full binary tree of height N .The pruning process leading to the outcome claimed in the proposition is basedon an iterative algorithm. The building block of the iteration is contained in Lemma12.3 below, with Lemma 12.2 supplying an easy but necessary intermediate step.130Lemma 12.2. Fix integers r ≥ 0 and C0 ≥ 1. A collection of cubes of cardinality≥ (2C0 + 1)d + 1 consisting of M-adic cubes of sidelength M−r and must contain atleast two cubes whose Euclidean separation is ≥ C0M−r.Proof. We first treat the case r = 0. The cube Q0 = [0, 2C0 + 1)d contains exactly(2C0 + 1)d subcubes of unit sidelength with vertices in Zd. The central subcube Qmaintains a minimum distance of C0 from the boundary of Q0. Rephrasing this aftera translation, any cube Q with vertices in Zd and of sidelength 1 admits at most(2C0 + 1)d similar cubes whose distance from itself is ≤ C0. The case of a generalr ≥ 0 follows by scaling Q0 by a factor of M−r.Lemma 12.3. Fix a constant integer C0 ≥ 1, an integer N0 ≥ (2C0 + 1)d and avertex v0 of the full Md-adic tree T ({1}× [0, 1)d;M). Let T[0] rooted at v0 be a subtreewith the property that every ray in T[0] splits at least N0 times. Then there exist aninteger K∗ = K∗(v0) ≥ 1 and a subtree T[1] of T[0] rooted at v0 and of height K∗ suchthat:(i) The root v0 has exactly two descendants v1 and v2 of height K∗ in T[1]. ThusT[1] has exactly one splitting vertex.(ii) The integerK∗ is the smallest with the property that dist(v1, v2) ≥ C0M−K∗−h(v0).In particular, dist(v1, v2) ≤ (C0 + 2√d)M−K∗−h(v0)+1.(iii) If T[0](vi) is the maximal subtree of T[0] rooted at vi then each ray in T[0](vi)splits at least N0 − (2C0 + 1)d times.Proof. Each ray in T[0] splits at least N0 times, so there exists a generation in thistree consisting of at least 2N0 vertices. Since 2N0  (2C0 + 1)d, let us define K0 tobe the smallest height in T[0] such that the number of vertices at that height exceeds(2C0 + 1)d. Choose K∗ to be the smallest integer with the property that there existvertices v1 and v2 of T[0] at height K∗ + h(v0) obeying the relation dist(v1, v2) ≥C0M−K∗−h(v0). By Lemma 12.2, this property holds at height K0, hence K∗ existsand is at most K0.131The subtree T1 of height K∗ rooted at v0 and generated by v1, v2 clearly obeysconditions (i) stated in Lemma 12.3. The lower bound on dist(v1, v2) required in part(ii) is built into the construction; see Figure 12.1. To obtain the upper bound, letv′i denote the parent of vi. It follows from the minimality of K∗ that dist(v′1, v′2) <C0M−K∗−h(v0)+1. Thus,dist(v1, v2) ≤ diam(v′1) + diam(v′2) + dist(v′1, v′2) ≤ (2√d+ C0)M−K∗−h(v0)+1.It remains to complete the proof of part (iii). Let us recall from the definition ofK0 that the number of elements of T[0] at height K∗ − 1 is ≤ (2C0 + 1)d. Thus anyray of T[0] rooted at v0 contains at most (2C0 + 1)d − 1 splitting vertices of height≤ K∗ − 2, since each splitting vertex of height ≤ K∗ − 2 gives rise to at least onenew element (different among themselves and distinct from the terminating vertexof the ray) at height K∗ − 1. Since every ray of T[0] contained at least N0 splittingvertices to begin with, at most (2C0 + 1)d of which may be lost by height K∗− 1, weare left with at least N0 − (2C0 + 1)d splitting vertices per ray rooted at vi, which isthe conclusion claimed in (iii).With the preliminary steps out of the way, we are ready to prove the mainproposition.Proof of Proposition 12.1. We know that split(T (Ω;M)) > (N+1)(2C0+1)d. Givenany N ≥ 1, we can therefore fix a subtree T of T (Ω;M) of infinite height in whichevery ray splits at least (N + 1)(2C0 + 1)d times. The pruning is executed on thesubtree T as follows.In the first step we apply Lemma 12.3 withT[0] = T , v0 = {1} × [0, 1)d and N0 = (N + 1)(2C0 + 1)d.This yields a subtree T[1] rooted at {1} × [0, 1)d of height K∗(v0) = i0 consistingof two vertices w1 and w2 at the bottom-most level. Let us denote by T (wi) themaximal subtree of T rooted at wi. By Lemma 12.3 any ray of T (wi) splits at least132va b?a ba1 b1a2 b2?a1 b1a2 b2v1v′2v2va bb2b1a2a1v1 v′2 v2Figure 12.1: An illustration of the procedure generating the forced Euclidean sepa-ration between the descendants v1 and v2 of v ∈ T , in R2 when M = 2.N(2C0 +1)d times. There is exactly one splitting vertex v in T[1], and its descendantsw1, w2 obey C0M−i0 ≤ dist(w1, w2) ≤ (C0 + 2√d)M−i0 . Set W1 := {w1, w2}.At the second step we invoke Lemma 12.3 twice, resetting the parameters in thatlemma to beT[0] = T (wi), v0 = wi, N0 = N(2C0 + 1)dfor i = 1, 2 respectively, and obtaining two subtrees as a consequence. Appendingthese two newly pruned subtrees of T (wi) to T[1] from the previous step, we arriveat a tree T[2] rooted at {1}× [0, 1)d of finite height but with rays of possibly variablelength, in which every ray splits exactly twice, and every splitting vertex has exactlytwo children. If v is a splitting vertex of T[2] not already considered in the first step,then v must either be equal to or a descendant of some w ∈ W1. Suppose that133w1(v) and w2(v) are the two descendants of w of maximal height in T[2]. Then part(ii) of Lemma 12.3 implies that (12.1) holds with h∗v = i0 + K∗(w). This verifiesthe requirements (i)-(iii) of Proposition 12.1 for N = 2. Let us denote by W2 thecollection of four vertices of maximal lineage in T[2] obtained at the conclusion of thisstep.In general at the end of the kth step we have a tree T[k] of finite height, butwith rays of potentially variable length, obeying the requirements (i)-(iii) for N = k.The collection of vertices of highest lineage in T[k] is termed Wk. We have that#(Wk) = 2k. The collection Wk can be decomposed asWk =⋃{Wk(v) : v is a splitting descendant of some w ∈Wk−1},where Wk(v) = {w1(v), w2(v)} consists of the two descendants of v that lie in Wk. Inthe (k+1)th step, Lemma 12.3 is applied 2k times in succession. In each application,the values of T[0], v0, N0 are reset toT[0] = T (w), v0 = w, N0 = (N − k + 1)(2C0 + 1)drespectively for some w ∈ Wk. The resulting tree T[k+1], obtained by appending the2k newly constructed trees to T[k] at the appropriate roots w, clearly obeys (ii) andalso (i) with N = k + 1. Part (iii) only needs to be verified for the splitting verticesv descended from some w ∈Wk, since the splitting vertices of older generations havebeen dealt with in previous steps. But this follows from part (ii) of Lemma 12.3,with h∗v = K∗(w) + h(w), which is a maximal height in the tree T[k+1].In view of the number of splitting vertices per ray in the original subtree T ,the process described above can be continued at least N steps. The tree T[N ] offinite height but variable ray lengths obtained at the conclusion of the Nth stepsatisfies the conditions (i)-(iii). We pick from every vertex of maximal lineage in T[N ]exactly one point of Ω, calling the resulting collection of 2N chosen points ΩN . Setδ := min{|ω − ω′| : ω, ω′ ∈ ΩN , ω 6= ω′} > 0. The rays in T[N ] are now extended asrays representing the points in ΩN (and hence without introducing any further splits)134to a uniform height J that satisfies M−J ≤ C−10 δ, thereby meeting the criterion inpart (iv).{1} × [0, 1)dw1v1w2v2height = i0Figure 12.2: An illustration of a pruned tree at the second step of pruning. Notethat, in general, we may have v1 = w1 and/or v2 = w2.12.2 Splitting and basic slope cubesThe pruned slope tree TJ(ΩN ;M) produced by Proposition 12.1 looks like an elon-gated version of the full binary tree of height N . Rays in this tree may have longsegments with no splits. However only the splitting vertices of TJ(ΩN ;M) and certainother vertices related to these are of central importance to the subsequent analysis.With this in mind and to aid in quantification later on, we introduce the class ofsplitting verticesG = G(ΩN) :=N⋃j=1Gj(ΩN), where for every 1 ≤ j ≤ N (12.2)Gj(ΩN) :={γ :there exists v ∈ ΩN such that γ is the jth splittingvertex on the ray identifying v in TJ(ΩN ;M)}. (12.3)135The vertices in Gj(ΩN) will be termed the jth splitting vertices. As dictated bythe pruning mechanism, such vertices γ may occur at different heights of the treeTJ(ΩN ;M), and hence could represent M -adic cubes of varying sizes. Thus theindex j, which encodes the number of splitting vertices on the ray leading up toand including γ, should not be confused with the height of γ in TJ(ΩN ;M). Givenγ ∈ G(ΩN), we writeν(γ) = j if γ ∈ Gj(ΩN), (12.4)and refer to ν(γ) as the splitting index of γ. Indeed N − ν(γ) is the splitting numberof γ with respect to TJ(ΩN ;M), defined as in (3.7). Note that G1(ΩN) consists of asingle element, namely the unique splitting vertex of T (ΩN ;M) of minimal height.In general #(Gj(ΩN)) = 2j−1; i.e., there are exactly 2j−1 splitting vertices of indexj. We declare GN+1(ΩN) ≡ ΩN .Another related quantity of importance is the one mentioned in part (iii) ofProposition 12.1. In view of its ubiquitous occurrence in the sequel, we set up thefollowing notation. For γ ∈ Gj(ΩN), 1 ≤ j ≤ N − 1, we denoteλ(γ) = λj(γ) := h∗γ defined as in Proposition 12.1. (12.5)Thus λj(γ) > h(γ) is the smallest height for which the Euclidean separation condition(12.1) can be ensured for the two descendants of γ. We refer to an element of{λj(γ) : γ ∈ Gj(ΩN)} as a jth fundamental height of ΩN . There could be at most2j−1 such heights. The collection of all fundamental heights will be denoted by R;it will play a vital role in the remainder of the article, specifically in the randomconstruction outlined in Chapter 14. The two descendants of γ ∈ Gj(ΩN) at heightλj(γ), are called the jth basic slope cubes. The entirety of jth basic slope cubes as γranges over Gj(ΩN) is termed Hj(ΩN). More precisely,Hj(ΩN) :={θ :there exists ω ∈ ΩN and γj ∈ Gj(ΩN)such that ω ∈ θ ( γj and h(θ) = λ(γj)}. (12.6)Note that every jth basic slope cube θ is either itself a (j + 1)th splitting vertex136γj+1 ∈ Gj+1(ΩN), or uniquely identifies such a vertex in the sense that there exists anon-splitting ray in the slope tree rooted at θ that terminates at γj+1. In either event,we say that γj+1 ∈ Gj+1(ΩN) is identified by θ ∈ Hj(ΩN). Since every γ ∈ Gj(ΩN)contributes exactly two cubes to Hj(ΩN), it follows that #(Hj(ΩN)) = 2j. We de-clare H0(ΩN) = G1(ΩN) and HN(ΩN) = ΩN .γ ∈ Gj(ΩN)γj+1θ1 θ2γ′j+1λj(γ)Figure 12.3: Two basic slope cubes θ1, θ2 ∈ Hj(ΩN) and their parent vertex γ ∈Gj(ΩN). Notice that γj+1 = θ1 and γ′j+1 are both members of Gj+1(ΩN).The following implication of the Euclidean separation condition (12.1) will beconvenient for later use.Corollary 12.4. Given a splitting vertex γ of TJ(ΩN), defineργ := sup{|a− b| : a ∈ γ1 ∩ ΩN , b ∈ γ2 ∩ ΩN}, (12.7)δγ := inf{|a− b| : a ∈ γ1 ∩ ΩN , b ∈ γ2 ∩ ΩN}, (12.8)where γ1 and γ2 are the two children of γ in TJ(ΩN ;M). Then, the two quantities ργand δγ are comparable; i.e., δγ ≤ ργ ≤ (1 + 2√dC−10 )δγ. Moreover, ργ ≤ C1M−λ(γ)for some constant C1 depending only on C0 and d.137Proof. Using part (iii) of Proposition 12.1 and the notation set up in (12.5), weobserve that γi ∩ ΩN ⊆ wi where wi is the only descendant of γi at height λ(γ), sothat δγ = dist(w1, w2) ≥ C0M−λ(γ). Let ai, bi be points in the closures of γi ∩ ΩN ,i = 1, 2 such that δγ = |a1 − a2|, ργ = |b1 − b2|. Thenργ = |b1 − b2| ≤ |a1 − b1| + |a2 − a1| + |a2 − b2|≤ |a2 − a1| + diam(w1) + diam(w2)≤ |a2 − a1| + 2√dM−λ(γ)≤ δγ + 2√dC−10 δγ,where the third inequality above follows from the fact that wi is itself a cube ofsidelength M−λ(γ). This shows that ργ and δγ are comparable. In order to obtain thestated upper bound on ργ, we observe that δγ ≤ (C0 + 2√d)M−λ(γ)+1 by (12.1).12.3 Binary representation of ΩNThe classes of basic slope cubes Hj(ΩN) allow us to represent each element in ΩN interms of a unique N -long binary sequence as follows. Since every splitting vertex ofTJ(ΩN) has exactly two children, one of them must be larger (or older) than the otherin the lexicographic ordering. Let us agree to call the older (respectively younger)child of a vertex v its 0th (respectively 1st) offspring. For 1 ≤ j ≤ N , we define abijective map Ψj : {0, 1}j → Hj(ΩN) inductively as follows. For j = 1,Ψ1(i) :={the unique element of H1(ΩN)descended from the ith child of γ1 ,(12.9)where i = 0, 1, and γ1 is the single element in H0(ΩN) = G1(ΩN). In general if Ψjhas been defined, then for ¯ ∈ {0, 1}j and i = 0, 1, we setΨj+1(¯, i) :={the unique element of Hj+1(ΩN)descended from the ith child of γj+1,(12.10)138where γj+1 is the unique element of Gj+1(ΩN) identified by Ψj(¯).The map ΨN provides the claimed bijection of {0, 1}N onto ΩN . In fact, thediscussion above yields the following stronger conclusion, the verification of which isstraightforward.Proposition 12.5. Let Hj(ΩN) be as in (12.6).(i) The collection of verticesH(ΩN) :=N⋃j=1{(θ1, · · · , θj)∣∣∣∃ ω ∈ ΩN , such that ω ∈ θk,θk ∈ Hk(ΩN), 1 ≤ k ≤ j}⋃{γ1} (12.11)is a tree rooted at γ1 ∈ H0(ΩN) of height N , in which (θ1, · · · , θj, θj+1) is avertex of height (j + 1) and a child of (θ1, · · · , θj). Every element θj ∈ Hj(ΩN)identifies a vertex (θ1, · · · , θj) of the jth generation in this tree.(ii) Let BN denote the full binary tree of height N , namely the tree TN([0, 1); 2).The map Ψ : BN → H(ΩN) defined byΨ(∅) = the unique element γ1 ∈ H0(ΩN),Ψ(¯) = Ψj(¯) if ¯ ∈ {0, 1}j, 1 ≤ j ≤ N,(12.12)with Ψj as in (12.10) is a tree isomorphism in the sense of Definition 3.1.Although we will not need to use it, an analogous argument shows that the classof splitting vertices G(ΩN) is isomorphic to BN−1.139Chapter 13Families of intersecting tubes,revisitedThe finite set of directions ΩN created in Proposition 12.1 forms the basis of theconstruction of Kakeya-type sets. Predictably, the sets of interest that verify theconclusion of Theorem 1.3 will be the union of a family of tubes, with each tubeassigned a slope from ΩN . Each tube is based on a suitably fine subcube of thed-dimensional root hyperplane, {0}× [0, 1)d. The tree depicting the root hyperplane,more precisely the full M -adic tree of dimension d and height J will be termed theroot tree.As in Chapter 6, for 0 ≤ k ≤ J , let Q(k) be the collection of all vertices of heightk in the root tree; i.e.,Q(k) :={Q : Q ∈ T ({0} × [0, 1)d;M), h(Q) = k}. (13.1)Geometrically, and in view of the discussion in Section 3.4, a member Q of Q(k) isan M -adic cube of sidelength M−k of the formQ = {0} ×d∏`=1[j`Mk, j`+1Mk), where (j1, j2, · · · jd) ∈ {0, 1, · · · ,Mk − 1}d, (13.2)140so that #(Q(k)) = Mkd. In view of the above, and for the purpose of distinguishingvertices of the root and the slope trees, a vertex in the root tree is termed a spatialcube. For reasons to be made clear in a moment, an element of Q(J) (i.e., a youngestvertex of the root tree) is of added significance and will be called a root cube.Given a fixed constant A0 ≥ 1, and for t ∈ Q(J), ω ∈ ΩN , we define a tube rootedat t with orientation ω to be the setPt,ω := ˜Qt + [0, 10A0]ω ={s+ rω : s ∈ ˜Qt, 0 ≤ r ≤ 10A0}. (13.3)Here ˜Qt denotes the cd-dilate of the cube t; i.e., the cube with the same centreas t but with cd times its sidelength, for a small positive constant cd specified inCorollary 7.2. For instance, the choice cd = d−2d will suffice. Thus Pt,ω is essentiallya (d + 1)-dimensional cylinder of constant length and with cubical cross-section ofsidelength cdM−J perpendicular to the x1-axis. An algorithm σ that assigns to everyroot t ∈ Q(J) a slope σ(t) ∈ ΩN produces, according to the prescription (13.3), afamily of tubes of cardinality MJd, and a corresponding setK(σ) = K(σ;N, J) :=⋃{Pt,σ(t) : t ∈ Q(J)}. (13.4)While this definition is quite general, in our applications the slope assignment mapσ will be chosen to be weakly sticky in the sense of Definition 13.6 and as a mappingbetween the trees representing roots and slopes respectively; specifically,σ : TJ({0} × [0, 1)d;M) → TJ(ΩN ;M).Random slope assignment algorithms will be prescribed in the next section, but fornow we record some properties of general sets of the form K(σ) generated by anarbitrary σ.14113.1 Tubes and a pointA crucial component of the proof of Proposition 3.7, amplified in Chapter 15, is toidentify when a given point x belongs to a union of tubes of the form (13.4). In ourapplications, the set K(σ) in (13.4) will be probabilistically generated by randomweakly sticky maps, and we will need to estimate the likelihood of such an inclusion.But many major ingredients of the argument pertain to general sets K(σ) generatedby an arbitrary weakly sticky σ. We discuss these features here.Directly analogous to Lemma 7.5 (a), we have the following lemma.Lemma 13.1. Let x ∈ Rd+1, A0 ≤ x1 ≤ 10A0. If the parameter C0 used in thepruning of the slope tree T (Ω;M) (see Proposition 12.1) is chosen sufficiently largerelative to the constant A0 in (13.3), then the following property holds: for anyt ∈ Q(J), there exists at most one v(t) ∈ ΩN such that x ∈ Pt,v(t).Proof. If there exist slopes v, v′ ∈ ΩN such that x ∈ Pt,v ∩ Pt,v′ , then the pointsx− x1v and x− x1v′ must both lie in t. In other words,|x1(v − v′)| = |(x− x1v) − (x− x1v′)| ≤√dM−J .Since x1 ≥ A0, this implies that |v − v′| ≤ A−10√dM−J , which is ≤ C02 M−J for achoice of C0 sufficiently large. Comparing with part (iv) of Proposition 12.1, we findthis is possible in ΩN only if v = v′.The lemma above motivates the following familiar definition: for x ∈ Rd+1 withA0 ≤ x1 ≤ 10A0,Poss(x) :={t ∈ Q(J) :there exists v(t) = v(t;x) ∈ ΩNsuch that x ∈ Pt,v(t)}. (13.5)Lemma 13.2. The set Poss(x) introduced in (13.5) can also be characterized asfollows:Poss(x) = {t ∈ Q(J) : t ∩ (x− x1ΩN) 6= ∅}. (13.6)142Thus Poss(x) is contained in an O(M−J)-neighborhood of an affine copy of ΩN inthe root hyperplane {0} × [0, 1)d.Proof. This lemma is just a restatement of Lemma 7.4, whose proof goes throughwithout alteration.The mappingv : Poss(x) → ΩN which sends t 7→ v(t) with x ∈ Pt,v(t) (13.7)is uniquely defined by Lemma 13.1. It captures for every t ∈ Poss(x) the “correctslope” that ensures that a tube rooted at t with that slope contains x. A purelydeterministic object driven by ΩN , this map has a certain structure that is criticalto the subsequent analysis. To formalize this property, let us recall the definitionsof Gj(ΩN) and Hj(ΩN) from (12.3) and (12.6). We denote for every ω ∈ ΩN and1 ≤ j ≤ N ,ηj(ω) := h(θ) where ω ⊆ θ ∈ Hj(ΩN). (13.8)In other words, ηj(ω) is the height of the jth basic slope cube on the ray identifyingω in TJ(ΩN ;M). We note that ηN(ω) ≡ J for all ω ∈ HN(ΩN) = ΩN .The quantity ηj is used to define the following objects:Nx := {Φj(t) : t ∈ Poss(x), 0 ≤ j ≤ N} , (13.9)Mx := {Θj(t) : t ∈ Poss(x), 0 ≤ j ≤ N} , where (13.10)Φj(t) :={0} × [0, 1)d for j = 0(Q∗1(t), · · · , Q∗j(t))for j ≥ 1, and(13.11)Θj(t) :={1} × [0, 1)d for j = 0(θ1(t), · · · , θj(t))for j ≥ 1.(13.12)Here for j ≥ 1, the cube Q∗j(t) is a cube in the root hyperplane containing t. Incontrast, θj(t) is a vertex in Hj(ΩN), hence a cube in {1} × [0, 1)d, containing thepoint v(t) ∈ ΩN . Furthermore, both cubes are located at the same height in their143respective trees and obey the defining propertiest ⊆ Q∗j(t), v(t) ∈ θj(t), and h(Q∗j(t)) = h(θj(t)) = ηj(v(t)). (13.13)We pause briefly to clarify the definitions (13.11) and (13.12) (see Figure 13.1).Given any t ∈ Poss(x), we pick on the ray identifying t the vertices that lie at thesame height as the basic slope cubes of v(t). The entries of the vector Φj(t) are thefirst j chosen vertices on this ray. On the other hand, Θj(t) consists of the first jbasic slope cubes containing v(t). The vectors ΦN(t) and ΘN(t) identify t and v(t)respectively. For reasons to emerge shortly in Lemma 13.4, we view the collectionNx as a tree, in which Φj(t) is a vertex of height j, and Φj+1(t) is a child of Φj(t).As we have already noted, the set Poss(x), and hence the youngest generation of Nx,contains all possible roots that could support tubes with directions in ΩN containingx. For an arbitrary σ, it is therefore natural to phrase a necessary criterion for theinclusion x ∈ K(σ) in terms of Nx. For this reason we choose to call Nx the referencetree, and its defining cubes Q∗j(t) as reference cubes. The collection Mx should bethought of as the “image” of Nx on the slope side, and hence a tree as well, withΘj(t) being a vertex of the jth generation and the parent of Θj+1(t). In fact, Mxis a subtree of H(ΩN) defined as in (12.11). In view of Proposition 12.5, any vertexΘj(t) of height j ≥ 1 in Mx is identified with the j-long binary sequence Ψ−1(Θj(t)).Given the constraints of our pruning mechanism in Proposition 12.1, the “cor-rect slope” map t 7→ v(t) need not be sticky as a mapping from TJ(Poss(x);M) toTJ(ΩN ;M). It does however possess a weak variant of the stickiness property thatwe specify in the next lemma. As we will see in Lemma 13.4, this milder substituteis able to achieve two goals that are of fundamental relevance to this study. First,it assigns a tree structure to Nx and Mx. Second, it is strong enough to lift v as asticky map from Nx →Mx.Lemma 13.3. There is a sufficiently large choice of the parameter C0 in Proposition12.1 for which the following conclusion holds. Let x ∈ Rd+1 with A0 ≤ x1 ≤ 10A0,t, t′ ∈ Poss(x) and u = D(t, t′). Set w = D(v(t), v(t′)), so that w ∈ G(ΩN), the class144ΦjΦn−1ΦntffffffΘjΘn−1Θnv(t)Figure 13.1: The pull-back mechanism used to define Nx, for M = d = 2.of splitting vertices defined in (12.2). Thenh(u) < λ(w), (13.14)with λ defined as in (12.5).Remark: If v defined in (13.7) was indeed a sticky map, one would have access tothe inequality h(u) ≤ h(w). We know however that λ(w) > h(w), and hence (13.14)should be viewed as a weak version of stickiness.Proof. If x ∈ Pt,v(t) ∩ Pt′,v(t′), then by the inequality (7.1) in Lemma 7.1,A0|v(t) − v(t′)| ≤ |x1||v(t) − v(t′)|≤ |cen(t′) − cen(t)| + 2√dM−J≤ 2√dM−h(u) + 2√dM−J ≤ 4√dM−h(u),and thus |v(t) − v(t′)| ≤ 4√dA−10 M−h(u). (13.15)145On the other hand, v(t) and v(t′) each lie in distinct children of w, which must bea splitting vertex of TJ(ΩN ;M). If w ∈ Gj(ΩN) and if γ, γ′ denote the (j + 1)thsplitting vertices descended from w, then each of γ and γ′ contains exactly one ofv(t) and v(t′). By Proposition 12.1(iii),|v(t) − v(t′)| ≥ dist(γ, γ′) ≥ C0M−λj(w). (13.16)Combining (13.15) and (13.16) we obtainC0M−λj(w) ≤ 4√dA−10 M−h(u).If the constant C0 is chosen larger than 4√dA−10 , then the inequality above implies(13.14), as claimed.Lemma 13.4. The collection of vertex tuples Nx, Mx defined in (13.9), (13.10) arewell-defined as trees rooted at {0} × [0, 1)d and {1} × [0, 1)d respectively, with theancestry relation as described in the discussion leading up to Lemma 13.3. Moreprecisely, the map v defined in (13.7) meets the following consistency requirements:(i) Let t, t′ ∈ Poss(x), u = D(t, t′). If the index j satisfies ηj(v(t)) ≤ h(u) then wealso have ηj(v(t′)) ≤ h(u), in which case Φj(t) = Φj(t′) and Θj(t) = Θj(t′).(ii) The map from Nx →Mx that sends Φj(t) 7→ Θj(t) is well-defined and sticky.Proof. Let γj(t) ∈ Gj(ΩN) denote the jth splitting vertex on the ray identifyingv(t). Then ηj(v(t)) = λj(γj(t)). If ηj(v(t)) = λj(γj(t)) ≤ h(u), then Lemma 13.3implies that ηj(v(t)) = λj(γj(t)) < λ(w), where w = D(v(t), v(t′)). Unravelling theimplication of this inequality, we see that the height of the first splitting descendantof γj(t) is strictly smaller than the corresponding quantity for w. Since both γj(t)and w are splitting vertices lying on the ray of v(t), this means that γj(t) is anancestor of w of strictly lesser height. In other words, w ⊆ γj+1(t). Since the raysfor v(t) and v(t′) agree up to and including height h(w), we conclude that their first(j + 1) splitting vertices are identical; i.e.,γk(t) = γk(t′) for k ≤ j + 1. (13.17)146Hence ηk(v(t)) = λk(γk(t)) = λk(γk(t′)) = ηk(v(t′)) for all such k, implying one ofthe desired conclusions in part (i). Sinceh(w) ≥ h(γj+1(t)) = h(γj+1(t′)) ≥ ηj(v(t)) = ηj(v(t′)),the vectors v(t) and v(t′) must agree at height ηj. Thus Θj(t) = Θj(t′). Of courseif ηj(v(t)) = ηj(v(t′)) ≤ h(u), then Φj(t) = Φj(t′). This completes the proof of thefirst part of the lemma.Part (ii) is essentially a restatement of the result in part (i). To ascertain thatthe map is well-defined we choose t, t′ ∈ Poss(x) with u = D(t, t′) and Φj(t) = Φj(t′)and aim to show that Θj(t) = Θj(t′). The hypothesis Φj(t) = Φj(t′) implies thatηj(v(t)) = ηj(v(t′)) ≤ h(u), and part (i) implies that the images match. Stickiness isa by-product of the definitions.Lemma 13.4 permits the unambiguous assignment of an “ideal image” (namelyan edge in Mx) to every edge of the tree Nx, in the following sense: if every edgein the ray leading up to ΦN(t) receives its ideal image, then x ∈ Pt,v(t). To makethis quantitatively precise, let us define the reference slope function κ as follows: forevery edge e in Nx joining the vertices Φj(t) to Φj+1(t), we define a binary counterκ(e) through the defining equationΨ−1 ◦ Θj+1(t) = (Ψ−1 ◦ Θj(t), κ(e)) (13.18)where Ψ is the tree isomorphism defined in Proposition 12.5. In other words, κ(e)is zero (respectively one) if and only if the ray identifying Θj+1(t) in TJ(ΩN) passesthrough the 0th (respectively 1st) child of the (j+ 1)th splitting vertex identified byΘj(t).Corollary 13.5. The reference slope function κ described in (13.18) is well-defined,and assigns to each edge of Nx a unique value of 0 or 1.Proof. If there exist t 6= t′ in Poss(x) such that the terminating vertex of e couldbe represented either as Φj+1(t) or as Φj+1(t′), then Lemma 13.4 guarantees that147Θk(t) = Θk(t′) for all k ≤ j + 1, proving that κ(e) given by (13.18) is a well-definedfunction on the edge set of Nx.The reader may find it helpful to visualize the edges of the reference tree Nxwith an overlay of model binary values assigned by κ, against which any other slopeassignment will be tested. This intuition is made precise below.13.2 Weakly sticky mapsMotivated by Lemma 13.3, we introduce a general notion of weak stickiness. This isa property that each random slope assignment σ prescribed in Chapter 14 will enjoy(see Lemma 14.1).Definition 13.6. Let σ be a height-preserving function that maps every full-lengthray of TJ({0} × [0, 1)d;M) to a full-length ray in TJ(ΩN ;M). We say that σ isweakly sticky if for any t, t′ ∈ Q(J), t 6= t′, one has the relation h(u) < λ(w), whereu = D(t, t′) and w = D(σ(t), σ(t′)).Given a fixed point x and a union of tubes K(σ) of the form (13.4) generated bya weakly sticky slope map σ, we obtain in Lemma 13.7 below a criterion governed bythe reference slope function κ for verifying whether x ∈ K(σ). Indeed for such σ, wecan define Nx(σ) and Mx(σ) akin to (13.9) and (13.10), but using the given slopemap t 7→ σ(t) instead of the naturally generated v given by (13.7). More precisely,we setNx(σ) := {Φj(t; σ) : t ∈ Poss(x), 0 ≤ j ≤ N}, (13.19)Mx(σ) := {Θj(t; σ) : t ∈ Poss(x), 0 ≤ j ≤ N}, (13.20)where for j ≥ 1, both Φj(t; σ) and Θj(t;σ) are j-long vectors whose ith componentsare M -adic cubes of identical size, containing t in the root hyperplane and σ(t) in theslope tree respectively. For Θj(t;σ), the ith entry is required to lie in Hi(ΩN), whichuniquely specifies both vectors. In light of the preceding results in this chapter, it is148not surprising that the collections (13.19) and (13.20) are trees and that σ extendsto a map between these trees.Lemma 13.7. The following conclusions hold:(i) The collections Nx(σ) and Mx(σ) as in (13.19) and (13.20) are well-defined astrees rooted respectively at {0} × [0, 1)d and {1} × [0, 1)d. The tuples Φj(t;σ)and Θj(t;σ) are deemed vertices of generation j, and parents of Φj+1(t;σ) andΘj+1(t;σ) respectively. The map Φj(t;σ) 7→ Θj(t; σ) from Nx(σ) → Mx(σ) iswell-defined and sticky.(ii) If e denotes the edge connecting Φj(t;σ) and Φj+1(t;σ) in Nx(σ), then thequantity ισ(e) defined byΨ−1 ◦ Θj+1(t; σ) = (Ψ−1 ◦ Θj(t; σ), ισ(e)) (13.21)gives rise to a well-defined binary function on the edge set of Nx(σ).(iii) If x ∈ K(σ), then there exists t ∈ Poss(x) such that ΘN(t; σ) = ΘN(t). Inparticular, this implies thatΦj(t; σ) = Φj(t) for all 1 ≤ j ≤ N, (13.22)and hence that Nx and Nx(σ) share a common ray R identifying t with thepropertyισ(e) = κ(e) for every edge e in R. (13.23)Proof. Not surprisingly, the proof of part (i) is a verbatim reproduction of the proofof Lemma 13.4 with v replaced by σ. The relation (13.14) which played a criticalrole in the proof of Lemma 13.4 is now ensured by the assumption of weak stickinessof σ. Part (ii) is an easy consequence of part (i) and follows exactly the same wayas Corollary 13.5 was deduced from Lemma 13.4. Finally, if x ∈ K(σ), then thereis some t ∈ Poss(x) such that σ(t) = v(t). Since the chain of basic slope cubescontaining any v ∈ ΩN is unique, this implies that ΘN(t;σ) = ΘN(t), and hence149Φj(t;σ) = Φj(t) for all 1 ≤ j ≤ N . The last equality says that t is identified bythe same sequence of vertices and hence the same ray in both Nx and Nx(σ). Ife1, e2, · · · eN are the successive edges in this ray, with ej+1 connecting Φj(t) withΦj+1(t), then a consequence of the definitions (13.18), (13.21) of κ and ισ is that(ισ(e1), · · · , ισ(eN)) = Ψ−1 ◦ σ(t) = Ψ−1 ◦ v(t) = (κ(e1), · · · , κ(eN)),where Ψ is the tree isomorphism defined in Proposition 12.5, and part (iii) follows.We end this section with a bound on the number of vertices of the reference tree ata given height, a result that will be useful for probability computations later. In viewof the characterization (13.6) of Poss(x) given in Lemma 13.2, and our constructionof ΩN , this is intuitively clear.Lemma 13.8. There exists a positive constant C depending on d and A0, but uniformin x ∈ [A0, A0 +1]×Rd, such that the number of vertices of height j in Nx is boundedabove by C2j.Proof. Let nj(x) denote the number of vertices of height j in Nx. In view of therelations (13.9) and (13.13) defining Nx, the cardinality nj(x) equals the number ofspatial cubes in the collection{Q∗j(t) : t ∈ Poss(x), t ⊆ Q∗j(t), h(Q∗j(t)) = ηj(v(t))}, (13.24)so we proceed to count the number of such cubes Q∗j(t). Let us recall from thedefinition (13.5) of Poss(x) that x ∈ Pt,v(t). This implies that x − x1v(t) ∈ t, andhence for θj(t) as in (13.13),|cen(Q∗j(t))−x+ x1cen(θj(t))|≤ |cen(Q∗j(t)) − cen(t)| + |x1||cen(θj(t)) − v(t)| + |cen(t) − x+ x1v(t)|≤√dM−ηj(v(t)) + (A0 + 1)√dM−ηj(v(t)) +√dM−J≤ 4A0√dM−ηj(v(t)).150Let us unravel the geometric implications of the inequality above. For a given θj(t)containing v(t), there are at most a constant number C(d,A0) of M -adic cubesof sidelength same as θj(t) (hence candidates for Q∗j(t)) whose centres are withindistance 4A0√dM−ηj(v(t)) of x − x1cen(θj(t)). On the other hand, each θj(t) ∈Hj(ΩN), and hence the total number of possible θj(t) as t ranges over Poss(x) is atmost #(Hj(ΩN)) = 2j, by Proposition 12.5. Since nj(x) is the cardinality of thecollection in (13.24), the observations above lead to the bound nj(x) = O(2j) asclaimed.151Chapter 14Random construction ofKakeya-type setsMotivated by the generalities laid out in the previous chapter, specifically Lemmas13.4 and 13.7, we now proceed to describe a randomized algorithm for generating aclass of weakly sticky slope assignments σ. Let us recall the class R of fundamentalheights of ΩN defined in (12.5) and the discussion thereafter.We start with a collection of independent and identically distributed Bernoulli(12)random variablesX := {XQ : Q ∈ Q(k), k ∈ R} , (14.1)with Q(k) defined as in (13.1). The collection X therefore assigns, for every fun-damental height k an independent binary random variable to every M -adic cube ofsidelength M−k in the root hyperplane. We use X as the randomization source forour construction.Let h0 denote the height of the single element θ0 ∈ G1(ΩN) = H0(ΩN), inother words, the first splitting vertex of TJ(ΩN ;M). We define σ(Q0) ≡ θ0 forall Q0 ∈ Q(h0). At the first step of the randomization process, each Q0 ∈ Q(h0) isdecomposed into subcubes Q1 of sidelength M−h1 where h1 = λ1(θ0) > h0. We callthese subcubes the first basic spatial cubes. Each first basic spatial cube Q1 receivesfrom the Bernoulli collection X defined in (14.1) a value of XQ1 , which is either zero152or one. Recalling from (12.9) thatΨ1(XQ1) ∈ H1(ΩN), and that h(Ψ1(XQ1)) = h1,we defineσ(Q1) = σX(Q1) = Ψ1(XQ1)for any first basic spatial cube Q1. Each element of H1(ΩN), and hence each σ(Q1),is either a second splitting vertex of ΩN or the identifier of one. If the root cube Q1already maps into a second splitting vertex under σ, no further action is needed forit in step one. Now, suppose there exists γ ∈ G2(ΩN) such that h(γ) > h1. Thenfor any cube Q1 for which Ψ1(XQ1) is the unique ancestor of γ at height h1, wedecompose Q1 into subcubes Q′1 of sidelength M−h(γ) and set σ(Q′1) = γ for all suchQ′1 ( Q1. Thus, at the end of the first step,(a) we have obtained a partition of the root hyperplane into first basic spatial cubes,and randomly assigned each such cube a first basic slope cube in H1(ΩN) of thesame height, namely λ1(θ0) = h1.(b) If the vertices in G2(ΩN) occur at different heights, then predicated on the ran-dom assignment in part (a) certain first basic spatial cubes could subdividefurther to generate a different partition of the root hyperplane, say {Q1(γ) : γ ∈G2(ΩN)}. Each cube Q′1 ∈ Q1(γ) is of height h(γ) and is mapped to γ. We willrefer to Q′1 as a spatial cube of second splitting height. Thus a first basic spa-tial cube is either itself a spatial cube of second splitting height, or is uniformlypartitioned into a disjoint union of such cubes.In general, the jth step of the construction generates a random and possibly non-uniform partition of the root hyperplane into spatial cubes Q′j of (j + 1)th splittingheight. Each Q′j is the terminal member of a descending chainQ′j ⊆ Qj ( Q′j−1 ⊆ Qj−1 ( · · · ( Q′1 ⊆ Q1, (14.2)where for every k ≤ j, Qk is a kth basic spatial cube, and Q′k is a spatial cube153γ ∈ Gj(ΩN)γj+1 = θ1 θ2γ˜j+1Q′j−1 ∈ Qj−1(γ)QjQ′jFigure 14.1: A pictorial representation of the basic slope and root cubes and atypical slope assignment. Vertices Qj for which XQj = 0 are indicated by a circleand assigned θ1; others are indicated by squares and assigned θ2. For the squaredvertices, a further slope assignment is made at a finer level.of (k + 1)th splitting height. Each Qk is mapped by σ to a kth basic slope cubein Hk(ΩN), whereas Q′k is mapped to a splitting vertex in Gk+1(ΩN). All suchassignments preserve heights and satisfy the following relation for a sequence ofcubes as in (14.2),σ(Q′j) ⊆ σ(Qj) ( σ(Q′j−1) ⊆ · · · ( σ(Q′1). (14.3)In this way, lineages of sequences of basic spatial cubes and spatial cubes of successivesplitting heights are preserved. However, notice that the full lineage of an arbitraryvertex in the root tree need not be preserved under σ.We may classify the spatial cubes at (j + 1)th splitting height as follows:Qj(γ) := {Q′j : σ(Q′j) = γ}, γ ∈ Gj+1(ΩN). (14.4)At the (j + 1)th step each Q′j from the collection Qj(γ) is decomposed into sub-cubes Qj+1 of height λj+1(γ) > h(γ). These are the (j + 1)th basic spatial cubes.154Each spatial cube Qj+1 is assigned the binary value XQj+1 from the Bernoulli collec-tion X in (14.1). Combined with the random assignments that the basic ancestorsof Qj+1 have received, this produces an image of Qj+1 under σ:σX(Qj+1) := Ψj+1(XQ1 , · · · , XQj+1) ∈ Hj+1(ΩN), Qj+1 ( · · · ( Q1. (14.5)Each σ(Qj+1) is the unique identifier of some γ ∈ Gj+2(ΩN). We decompose Qj+1into subcubes Q′j+1 of height h(γ) (in some cases no further decomposition may beneeded) and set σ(Q′j+1) = γ. This results in a newer and finer partition of theroot hyperplane into spatial cubes Q′j+1 of (j + 1)th splitting height, producing ananalogue of (14.4) for the (j + 2)th step and allowing us to carry the inductionforward.Continuing the procedure described above for N steps, we obtain a decompositionof the root hyperplane into a family of basic cubes of order N , each of which is ofsidelength M−J , and hence is by definition a root cube. Every such cube t = QN(t)is contained in a unique chain of basic spatial cubes of lower order:t = QN(t) ( QN−1(t) ( · · ·Q2(t) ( Q1(t) (14.6)and is assigned a slope σX(t) = ΨN(XQ1 , · · · , XQN ) in HN(ΩN) = ΩN . We willshortly expand on further structural properties of the slope map t 7→ σX(t), but firstobserve that it gives rise to a random setKN(X) := K(σX;N, J) (14.7)according to the prescription (13.4).14.1 Features of the constructionWe pause briefly to summarize the important features of the construction above:- Randomization only occurs for cubes in the root hyperplane that correspond to155the fundamental heights, though all cubes of a given fundamental height need notreceive a random assignment.- The only cubes that receive a random binary assignment from X are by definitionthe basic spatial cubes. Unlike the basic slope cubes that constitute Hj(ΩN), a basicspatial cube Qj is a random quantity. For instance, the size of a jth basic spatialcube Qj always ranges in the set {h(θ) : θ ∈ Hj(ΩN)} ⊆ R, but the exact valueof the size depends on the binary assignment XQ1 , · · · , XQj−1 received by its basicancestors. Similarly, a spatial cube Q′j of jth splitting height is random, though ofcourse a splitting vertex in Gj(ΩN) is not.- On the other hand, the random variable XQj that a basic spatial cube Qj receivesis independent of all random variables used in previous or concurrent steps of theprocess, by virtue of our choice of (14.1). In other words,The collection of random variables{XQj : Qj basic}is independent. (14.8)This fact is vital in computing slope assignment probabilities in Chapters 15 and16.- Thus far, σ has been prescribed only for basic cubes and their subcubes of splittingheights. Having achieved this, it is not difficult to extend σ as a weakly sticky mapbetween the root tree and the slope tree. We address this in the next lemma.Lemma 14.1. For every realization of X, there exists a weakly sticky mapσX : TJ({0} × [0, 1)d;M) → TJ(ΩN ;M)in the sense of Definition 13.6 that agrees with the slope assignment algorithm pre-scribed in (14.5).Proof. The prescriptions made in (14.5) show that σX assigns a full ray in TJ({0} ×[0, 1)d;M) to one in TJ(ΩN ;M). We aim to show that h(u) < λ(w) for u = D(t, t′)and w = D(σX(t), σX(t′)). Since w is by definition a splitting vertex in TJ(ΩN ;M),156let j denote the index such that w ∈ Gj(ΩN). Indeed, if h(u) ≥ λ(w), let Q be thecommon ancestor of t and t′ at height λ(w). Then Q is a jth basic spatial cube,which is mapped under σX to a jth basic slope cube θ. Thus θ is a common ancestorof σX(t) and σX(t′) at height λ(w) > h(w), contradicting the definition of w.14.2 Theorem 1.3 revisitedWe will now invest our efforts into proving that with positive probability the setsKN(X) just created in (14.7) are of Kakeya type.Proposition 14.2. There exist positive absolute constants c = c(d,M) and C =C(d,M) obeying the property described below. For every N ≥ 1 and ΩN as inProposition 12.1, the random set KN(X) defined in (14.7) satisfies the followinginequalities:Pr({X : |KN(X) ∩ [0, 1] × Rd| ≥ c√logNN})≥ 34, (14.9)EX∣∣KN(X) ∩ [A0, A0 + 1] × Rd∣∣ ≤ CN. (14.10)The proof of the proposition will occupy the remainder of the main document,with the estimates (14.10) and (14.9) established in Chapters 15 and 19 respectively.Before launching into them, let us observe that these two estimates combine to gener-ate the Kakeya-type set whose existence is claimed in Theorem 1.3 and subsequentlyreformulated in Proposition 3.7.Corollary 14.3. Given Proposition 14.2, the statement of Proposition 3.7 follows.Specifically, for every N ≥ 1 there exists a realization of X for which the union oftubes defined byEN := KN(X) ∩ [A0, A0 + 1] × Rd obeys|E∗N(2A0 + 1)||EN |−→N→∞∞. (14.11)In other words, Ω admits Kakeya-type sets.157Proof. The proof is identical to that of Proposition 6.2, where we set aN = c√lognNand bN = CN .158Chapter 15Proof of the upper bound (14.10)Proposition 15.1. There exists a positive constant C possibly depending on d andMbut uniform in x ∈ [A0, A0+1]×Rd such that the probability Pr(x) := Pr(x ∈ KN(X))obeys the estimatePr(x) ≤ CN. (15.1)As a consequence, (14.10) holds.Proof. The proof of (15.1) is a consequence of the three lemmas stated and provedbelow in this chapter. In Lemma 15.3 and following the direction laid out in [4, 3],we establish that Pr(x) is bounded above by the probability that the reference treeNx survives a Bernoulli(12) percolation, as described in Chapter 4. The details of thespecific percolation criterion that permit this correspondence are described in Lemma15.2. Using general facts about percolation collected in Chapter 4 and informationon Nx observed in Chapter 13, we compute in Lemma 15.4 a bound on the survivalprobability that is uniform in x to obtain the claimed estimate (15.1).Given (15.1), the upper bound in (14.10) follows easily. Since ΩN ⊆ {1}× [0, 1)d,any tube, and hence KN(X), is contained in the compact set [0, 10A0]d+1. ThusKN(X) ∩ [A0, A0 + 1] × Rd = KN(X) ∩ [A0, A0 + 1] × [0, 10A0]d,159and henceEX∣∣KN(X) ∩ [A0, A0 + 1] × Rd∣∣ = EX∫[A0,A0+1]×[0,10A0]d1KN (X)(x) dx=∫[A0,A0+1]×[0,10A0]dEX(1KN (X)(x))dx=∫[A0,A0+1]×[0,10A0]dPr(x) dx≤ CN,completing the proof.Much of the groundwork for Lemma 15.3 has already been established in Section13.1. In particular, let us recall the definition of the reference tree Nx and referencecubes Q∗j(t) from (13.9), (13.11), and (13.13). We will also need the reference slopefunction κ as in (13.18) defined on the edges of Nx. Motivated by Lemma 13.7(iii),we define a random variable for each edge of Nx:Ye = Ye(X) :=1 if XQ∗j+1(t) = κ(e),0 otherwise,(15.2)where as usual e denotes the edge in Nx joining Φj(t) and Φj+1(t). As describedin Chapter 4, we use Ye to determine whether to retain or to remove the edge e inNx, the value zero corresponding to removal. We emphasize that a reference cubeQ∗j+1(t) is a deterministic vertex of the tree representing the root hyperplane, andneed not in general coincide with the (j + 1)th basic spatial cube Qj+1(t) describedin (14.6). The important point, as we will see in Lemma 15.3, is that if x ∈ KN(X),then these two cubes do match for some t and for every j.Lemma 15.2. The retention-removal criterion described in (15.2) gives rise to awell-defined Bernoulli(12) percolation on Nx.Proof. Since Q∗j+1(t) identifies the terminating vertex of the edge e, any two rep-resentations Φj+1(t) = Φj+1(t′) of this vertex gives rise to Q∗j+1(t) = Q∗j+1(t′). So160XQ∗j+1(t) is consistently defined on the edges. We have already seen in Corollary 13.5that κ is a well-defined function on the edge set of Nx, hence so is Ye. The probabilitythat Ye equals one is clearly 1/2 since it is given by the Bernoulli(12) random variableXQ∗j+1(t). Finally, any two distinct edges e and e′ must have distinct terminatingvertices, and therefore end in distinct reference cubes. The random variable assign-ments for such cubes are independent by our assumption on X. Hence the events ofretention and removal are independent for different edges, and the result follows.Lemma 15.3. Let x be a point in [A0, A0 + 1]×Rd. If x ∈ KN(X), then there is atleast one ray of full length in Nx all of whose edges are retained after the percolationdescribed by Ye(X). As a result, the probability Pr(x) defined in Proposition 15.1admits the boundPr(x) ≤ p∗(x), (15.3)where p∗(x) denotes the survival probability of Nx under the Bernoulli(12) percolationgiven in (15.2).Proof. If x ∈ KN(X), then by Lemma 13.7(iii) there exists t ∈ Poss(x) such that theray identifying t is common to Nx and Nx(σX). Restating (13.22), this means thatΦj(t) = Φj(t;σ) for all 1 ≤ j ≤ N . But the left hand side of the preceding equalityidentifies the (deterministic) jth reference cube containing t, whereas the right handside represents the (random) jth basic spatial cube containing t. In other words, wefind that Qj(t;X) = Q∗j(t) for all 1 ≤ j ≤ N , and henceισ(e) = XQj+1(t;X) = XQ∗j+1(t).Combined with (15.2) and (13.23), this implies the existence of an entire ray in Nx(namely the one identifying t) that survives the percolation given by Ye. Summariz-ing, we obtain that{X : x ∈ KN(X)} ⊆{X :Nx survives the Bernoulli(1/2)percolation dictated by Ye(X)},from which (15.3) follows.161Lemma 15.4. There is a positive constant C that is uniform in x ∈ [A0, A0+1]×Rdsuch that the survival probability p∗(x) of Nx under Bernoulli(12) percolation is ≤CN .Proof. In view of Corollary 4.2, p∗(x) is bounded above by[N∑j=12jnj(x)]−1where nj(x) = number of vertices in Nx of height j.But Lemma 13.8 gives that nj(x) ≤ C2j, which leads to the stated bound.162Chapter 16Probability estimates for slopeassignmentsWe now turn to (14.9), where we need to establish that with high probability, thevolume of space close to the root hyperplane is much more widely populated by therandom set KN(X) than away from it. As we have already seen in Section 9 duringthe Cantor directions case, the proof requires detailed knowledge of the probabilitythat a given subset of root cubes receives prescribed slope assignments. We establishthe necessary probabilistic estimates in this section for easy reference in the proof of(14.9), which is presented in Chapter 19.16.1 A general ruleTo get started, let us recall from Chapter 14 that a slope assigned to a root is notcompletely arbitrary and has to obey the requirement of weak stickiness. The defi-nition below, introduced to avoid vacuous root-slope combinations, draws attentionto this constraint.Definition 16.1. Let A be a collection of root cubes and ΓA = {α(t) : t ∈ A} ⊆ ΩN163a collection of slopes indexed by A. We say that the collection of root-slope pairs{(t, α(t)) : t ∈ A ⊆ Q(J), α(t) ∈ ΓA ⊆ ΩN} (16.1)is sticky-admissible if there exists a realization of X as in (14.1) for which the weaklysticky map σX described in Chapter 14 has the property thatσX(t) = α(t) for all t ∈ A. (16.2)Given a sticky-admissible collection (16.1), we first prescribe a general algorithmfor computing the probability of the event (16.2). Preparatory to stating the result,let us define two collections consisting of tuples of vertices from the root tree andthe slope tree respectively:N(A;α) := {Φj(t;α) : t ∈ A, 0 ≤ j ≤ N}, (16.3)M(A;α) := {Θj(t;α) : t ∈ A, 0 ≤ j ≤ N}. (16.4)These objects are analogous to the trees (13.19) and (13.20) introduced earlier, withthe usual interpretation of Φj(t;α) and Θj(t;α) following those definitions. Namely,for j ≥ 1, the element Θj(t;α) is a vector with j entries, whose ith componentrepresents the ith basic slope cube in TJ(ΩN) containing α(t). The vector Φj(t;α) isalso a j-long sequence. Its ith entry represents the unique cube containing t locatedat the same height as the ith entry of Θj(t;α). This common height is ηi(α(t))defined as in (13.8). Not surprisingly, for a choice A and α that gives rise to asticky-admissible collection (16.1), the collections N(A;α) and M(A;α) are indeedtrees (with the 0th generations removed) that contain the information required forcomputing the probability of the event (16.2). This is the content of Lemma 16.2below, which forms the computational framework for all the probability estimates inthis section.Lemma 16.2. Let A ⊆ Q(J) and ΓA = {α(t) : t ∈ A} ⊆ ΩN be sets for which thecollection given in (16.1) is sticky-admissible. Then the following conclusions hold.164(i) The collections N(A;α) and M(A;α) defined in (16.3) and (16.4) are well-defined trees in which Φj(t;α) and Θj(t;α) are deemed vertices of height j, andparents of Φj+1(t;α) and Θj+1(t;α) respectively.(ii) If n(A;α) denotes the total number of vertices in N(A;α) not counting the root,thenPr(σX(t) = α(t) for all t ∈ A) = 2−n(A;α). (16.5)Proof. The proof of the first claim follows the same line of reasoning as in Lemmas13.3 and 13.7 and is hence omitted. We turn to the proof of (16.5). Let us writeΦj(t;α) =(Q∗1(t;α), · · · , Q∗j(t;α))and Θj(t;α) =(θ1(t;α), · · · , θj(t;α)).(16.6)In order to describe the event of interest, we need to recall from (14.6) the definitionof basic spatial cubes Qj(t) containing t, their role in the random construction asexplained in Chapter 14, and also the definition of the maps Ψj and Ψ from (12.10)and Proposition 12.5. Putting these together we find that{σX(t) = α(t) for all t ∈ A}={σX(Qj(t)) = θj(t;α) for all 1 ≤ j ≤ N and all t ∈ A}={ΨN(XQ1(t), · · · , XQN (t)) = α(t) for all t ∈ A}=N⋂j=1⋂t∈A{XQj(t) = pij ◦ Ψ−1 ◦ α(t)}=N⋂j=1⋂t∈A{Qj(t) = Q∗j(t;α) and XQ∗j (t;α) = pij ◦ Ψ−1 ◦ α(t)}. (16.7)Here pij denotes the projection onto the jth component of an input sequence. In thefirst two steps of the string of equations above, we have used the definition (14.5) of σand its weak stickiness as ensured by Lemma 14.1. To justify the last step we observethat Q1(t) = Q∗1(t;α) is non-random; further if it is given that Q`(t) = Q∗`(t;α) for165all ` ≤ j, then the additional requirementXQj(t) = pij ◦ Ψ−1 ◦ α(t) implies Qj+1(t) = Q∗j+1(t;α),leading to the conclusion in (16.7). By virtue of our assumption of sticky-admissibility,the event described above is of positive probability; in particular the value assignmentto the random variables in X as prescribed in (16.7) is consistent; i.e., for t 6= t′,pij ◦ Ψ−1 ◦ α(t) = pij ◦ Ψ−1 ◦ α(t′) whenever Q∗j(t;α) = Q∗j(t′;α).In view of our assumption (14.1) on the distribution of X, the probability of theevent in (16.7) is half raised to a power that equals the number of distinct cubes inthe collection {Q∗j(t;α); 1 ≤ j ≤ N, t ∈ A}, in other words n(A;α).16.2 Root configurationsApplication of Lemma 16.2 requires explicit knowledge of the structure of the treesN(A;α) and M(A;α), from which n(A;α) can be computed. These objects dependin turn on the trees depicting A and ΓA. We now proceed to compute n(A;α) insome simple situations where #(A) ≤ 4. On one hand, the small size of A permitsthe classification of possible root configurations into relatively few categories, eachof which gives rise to a specific n(A;α). On the other hand, these cases cover all theprobabilistic estimates that we will need in Chapter 19.While each root configuration requires distinct consideration, it is recommendedthat the reader focus on the cases when #(A) = 2, and when #(A) = 4 with the fourroots in what we call a type 1 configuration (see Definition 16.7). These cases containmany of the main ideas needed to push through the proof of the lower bound on thesize of a typical KN(X) claimed in (14.9), Proposition 14.2. A thorough treatment ofall distinct cases when #(A) ≤ 4 is needed to completely establish Proposition 14.2,but focusing on the two recommended cases should make the arguments far easierto absorb upon a first reading. When #(A) = 2 in particular, the reader may focus166attention on Lemmas 16.3, 17.3, 18.1 and 18.2, and the application of these lemmasin the proof of Proposition 19.1. The treatment of the case of four distinct roots intype 1 configuration has been carried out on Lemmas 16.8, 17.6, 18.1 and 18.2, withthe application of these lemmas occurring in the proof of Proposition 19.2, for whichthis is the generic case.16.3 NotationThroughout this chapter the following notation will be used, in conjunction with theterminology of root hyperplane, root tree and root cube already set up in Chapter13, page 140. Since any vertex Φj(t;α) in N(A;α) is uniquely identified by its lastcomponent Q∗j(t;α) defined as in (16.6), we writeΦj(t;α) ∼= Q∗j(t;α), (16.8)often opting to describe the left hand side by the right. In particular if j = N , thenΦN(t;α) ∼= Q∗N(t;α) = t, in which case the latter notation is used instead of the(more cumbersome) former.Given a vertex u in the root tree, a vertex ω ∈ TJ(ΩN) and a positive integer ksuch that k ≤ h(u) ≤ λ(w), we also defineθ(ω, k) := the basic slope cube containing ω of maximal height ≤ k, and (16.9)µ(ω, k) := j if θ(ω, k) ∈ Hj(ΩN)= number of basic slope cubes of height ≤ k that contain ω, and(16.10)Qu[ω, k] := ancestor of u in the root tree at height h(θ(ω, k)). (16.11)Figure 16.1 on page 168 depicts these quantities. If ω′ ⊆ ω and/or u′ ⊆ u, then itfollows from the definitions above thatθ(ω, k) = θ(ω′, k), µ(ω, k) = µ(ω′, k), and167Qu[ω, k] = Qu′ [ω, k] = Qu[ω′, k] = Qu′ [ω′, k].These facts will be frequently used in the sequel without further reference.ωj ∈ Gj(ΩN)θ(ω, k1) = θ(ω, k2) ∈ Hj(ΩN)θ(ω, k3) ∈ Hj+1(ΩN)height = k2height = k1height = k3ω ∈ TJ(ΩN)Figure 16.1: Given ω ∈ ΩN and a set of heights ki, i = 1, 2, 3, the basic slope cubesθ(ω, ki) are identified. Here µ(ω, k1) = µ(ω, k2) = j and µ(ω, k3) = j+1. All verticesdepicting basic slope cubes are circled.16.4 The case of two rootsWe start with the simplest case when A consists of two root cubes.Lemma 16.3. Let A = {t1, t2} be two distinct root cubes and ΓA = {α(t1) =v1, α(t2) = v2} ⊆ ΩN be a subset of (not necessarily distinct) slopes such that{(t1, v1), (t2, v2)} is sticky-admissible. If u = D(t1, t2), ω = D(v1, v2) and k = h(u),168then k < λ(ω), andPr(σ(t1) = v1, σ(t2) = v2)=(12)2N−µ(ω,k). (16.12)Proof. Since there exists a weakly sticky map σ such that σ(ti) = vi for i = 1, 2, wesee thatλ(ω) = λ(D(v1, v2))= λ(D(σ(t1), σ(t2)))≥ h(D(t1, t2))= h(u) = k. (16.13)In order to establish (16.12) we invoke Lemma 16.2. The tree N(A;α) consists oftwo rays terminating at ΦN(t1;α) ∼= t1 and ΦN(t2;α) ∼= t2 respectively, according tothe notational rule prescribed in (16.8). Letting uN = DN(t1, t2) denote the youngestcommon ancestor of t1 and t2 in N(A;α), we observe that uN ∼= Qu[ω, k], with Qu[ω, k]defined as in (16.11). Thus uN lies at height µ(ω, k) in N(A;α). This allows us tocompute n(A;α) as follows: n(A;α) = µ(ω, k) + 2(N − µ(ω, k)) = 2N − µ(ω, k).16.5 The case of three rootsNext we turn to the slightly more complex event where three distinct root cubesreceive prescribed slopes. Here for the first time we observe the dependence of slopeassignment probabilities on configuration types of the roots.Definition 16.4. Let t1, t2, t′2 be three distinct root cubes. We say that the orderedtuple I = {(t1, t2); (t1, t′2)} withu = D(t1, t2), u′ = D(t1, t′2), u′ ⊆ u (16.14)is in type 1 configuration if exactly one of the following conditions hold:(a) u′ ( u, or(b) u = u′ = D(t2, t′2).169A tuple I that obeys (16.14) but is not of type 1 is said to be of type 2. Thus forI of type 2, one must have u = u′ and additionally t = D(t2, t′2) satisfies t ( u.If I = {(t1, t2); (t1, t′2)} with the same definitions of u and u′ does not meet thecontainment relation required by (16.14), i.e., if u ( u′, then we declare I to be ofthe same type as I′ = {(t1, t′2); (t1, t2)}.The different structural possibilities are shown in Figure 16.2.3 Point ConfigurationsType 1 Type 2uu′t1 t′2 t2u = u′t′2t1 t2u = u′t1 t2 t′2Figure 16.2: All possible configurations of three distinct root cubes.As in Lemma 16.3, the quantity µ defined in (16.10) when evaluated at certainvertices of the slope tree dictated by A = {t1, t2, t′2} provides the value of n(A;α)necessary for estimating the probability in (16.5).Lemma 16.5. Let A = {t1, t2, t′2} be three distinct root cubes such that the orderedtuple I = {(t1, t2); (t1, t′2)} obeys (16.14) and is of type 1. Setk = h(u), k′ = h(u′).Suppose that ΓA = {α(t1) = v1, α(t2) = v2, α(t′2) = v′2} ⊆ ΩN is a subset of(not necessarily distinct) directions such that the collection {(t1, v1); (t2, v2); (t′2, v′2)}is sticky-admissible. Then the vertices defined byω = D(v1, v2), ω′ = D(v1, v′2)170must satisfy the height relationsk ≤ λ(ω), k′ ≤ λ(ω′) (16.15)and the following equality holds:Pr(σ(t1) = v1, σ(t2) = v2, σ(t′2) = v′2)=(12)3N−µ(ω,k)−µ(ω′,k′).Proof. The inequalities in (16.15) are proved exactly as in Lemma 16.3; we omitthese. The probability is again computed using Lemma 16.2, via counting n(A;α).The tree N = N(A;α) now consists of three rays, terminating at ΦN(t1;α), ΦN(t2;α)and ΦN(t′2;α), which are identified with t1, t2 and t′2 respectively. Let us recall fromthe proof of Lemma 16.3 that uN = DN(t1, t2) denotes the M -adic cube specifyingthe youngest common ancestor of t1 and t2 in N(A;α). The vertex u′N = DN(t1, t′2)is defined similarly. Then using the notation (16.8),u′N ∼= Qu′ [ω′, k′] = Qt1 [v1, k′], and uN ∼= Qu[ω, k] = Qt1 [v1, k]. (16.16)Since k ≤ k′, it follows from (16.16) above that u′N ⊆ uN. If hN(·) denotes the heightof a vertex within the tree N(A;α), then (16.16) also yieldshN(uN)= µ(ω, k) and hN(u′N)= µ(ω′, k′), so that µ(ω, k) ≤ µ(ω′, k′).Using these relations and referring to Figure 16.2, we compute n(A;α) as follows,n(A;α) = hN(uN)+[N − hN(uN)]︸ ︷︷ ︸vertices on the ray of t2 in N+[hN(u′N)− hN(uN)]︸ ︷︷ ︸vertices between uN and u′N+ 2[N − hN(u′N)]︸ ︷︷ ︸vertices below u′N= µ(ω, k) +[N − µ(ω, k)]+[µ(ω′, k′) − µ(ω, k)]+ 2[N − µ(ω′, k′)]= 3N − µ(ω, k) − µ(ω′, k′),which leads to the desired probability estimate by Lemma 16.2.171Lemma 16.6. Let A = {t1, t2, t′2} be three distinct root cubes such that the orderedtuple I = {(t1, t2); (t1, t′2)} obeys (16.14) and is of type 2. Setk = h(u) = h(u′), and ` = h(t) where t = D(t2, t′2) ( u = u′.If {(t1, v1); (t2, v2); (t′2, v′2)} is a sticky-admissible collection, then the verticesω = D(v1, v2), ω′ = D(v1, v′2), ϑ = D(v2, v′2)must satisfy the relationsk ≤ min{λ(ω), λ(ω′)}, ` ≤ λ(ϑ), µ(ω, k) = µ(ω′, k), (16.17)and the following equality holds:Pr(σ(t1) = v1, σ(t2) = v2, σ(t′2) = v′2)=(12)3N−µ(ω,k)−µ(ϑ,`). (16.18)Proof. The first two inequalities in (16.17) are consequences of weak stickiness, sincethere exists a weakly sticky map σ that assigns σ(t1) = v1, σ(t2) = v2, σ(t′2) = v′2.Thus the first inequality in (16.17) is proved as in (16.13), while the second one alsofollows a similar route:λ(ϑ) = λ(D(v2, v′2)) = λ(D(σ(t2), σ(t′2)))≥ h(D(t2, t′2)) = h(t) = `.For the last identity in (16.17), we observe that both ω and ω′ lie on the ray identi-fying v1. Thus θ(ω, k) = θ(ω′, k) and hence µ(ω, k) = µ(ω′, k) by the first inequalityin (16.17).We now turn to the counting of n(A;α), which leads to the probability estimate(16.18) via Lemma 16.2. Using the notation introduced in the proof of Lemma 16.5,the pairwise youngest common ancestors of the last generation vertices in N(A;α)172are seen to satisfy the following:uN = DN(t1, t2) ∼= Qu[ω, k] = Qt2 [v2, k],tN = DN(t2, t′2) ∼= Qt[ϑ, `] = Qt2 [v2, `].Since the type of I guarantees that k < `, the relations above implytN ⊆ uN and hence hN(uN) = µ(ω, k) ≤ hN(tN) = µ(ϑ, `).This enables us to compute, with the aid of Figure 16.2,n(A;α) = µ(ω, k) +[N − µ(ω, k)]︸ ︷︷ ︸vertices on the ray of t1 in N+[µ(ϑ, `) − µ(ω, k)]︸ ︷︷ ︸vertices between uN and tN+ 2[N − µ(ϑ, `)]︸ ︷︷ ︸vertices below tN= 3N − µ(ω, k) − µ(ϑ, `).This is the exponent claimed in (16.18).16.6 The case of four rootsFinally we turn our attention to four point root configurations. Depending on therelative positions of root cubes within the root tree, we can classify the configurationtypes as follows. Let I = {(t1, t2); (t′1, t′2)} be an ordered tuple of four distinct rootcubes, for whichu = D(t1, t2) and u′ = D(t′1, t′2) obey h(u) ≤ h(u′). (16.19)Then exactly one of the following conditions must hold:u ∩ u′ = ∅, (16.20)u = u′ = D(ti, t′j) for all i, j = 1, 2, (16.21)u′ ( u, (16.22)173u = u′, and ∃ indices 1 ≤ i, j ≤ 2 such that D(ti, t′j) ( u. (16.23)Definition 16.7. For an ordered tuple I = {(t1, t2); (t′1, t′2)} of four distinct rootcubes meeting the requirement of (16.19), we say that I is of(a) type 1 if exactly one of (16.20) or (16.21) holds,(b) type 2 if (16.22) holds, and(c) type 3 if (16.23) holds.If I does not meet the height relation in (16.19), then I′ = {(t′1, t′2); (t1, t2)} does, andthe type of I is said to be the same as that of I′.Several different structural possibilities for the root quadruple exist within theconfines of a single type, excluding permutations within and between the pairs {t1, t2}and {t′1, t′2}. These have been listed in Figure 16.3. We note in passing that thetype definition above is slightly different from the Cantor case of Chapter 9. Here,the main motivation for the nomenclature is the classification of the unconditionalprobabilities of slope assignment as exemplified in (16.5), whereas in Chapter 9 asimpler analysis involving conditional probabilities only was possible.We now proceed to analyze how the configuration types affect the slope assign-ment probabilities.Lemma 16.8. Let A = {t1, t2, t′1, t′2} be a collection of four distinct root cubes suchthat I = {(t1, t2); (t′1, t′2)} obeys (16.19) and is of type 1. Let ΓA = {v1, v2, v′1, v′2} ={α(ti) = vi, α(t′i) = v′i, i = 1, 2} ⊆ ΩN be a choice of slopes such that the collection{(ti, vi); (t′i, v′i); i = 1, 2} is sticky-admissible. Setz = D(u, u′), k = h(u), k′ = h(u′), ` = h(z),so thatu, u′ ( z, and hence ` < k ≤ k′ if (16.20) holds,u = u′ = z, and hence ` = k = k′ if (16.21) holds.174Type 1Configurations u u′t1 t2 t′1 t′2u = u′t1 t′1 t2 t′2Type 2Configurationsuu′t2t1 t′1 t′2uu′t1 t′1 t′2 t2uu′t1 t2 t′1 t′2Type 3Configurationsu = u′t1 t′1 t2 t′2u = u′t1 t2 t′2 t′1Figure 16.3: Configurations of four root cubes, up to permutations.Then the verticesω = D(v1, v2), ω′ = D(v′1, v′2), v = D(ω, ω′),175must satisfy k ≤ λ(ω), k′ ≤ λ(ω′) and ` ≤ λ(v), and the following equality holds:Pr(σ(ti) = vi, σ(t′i) = v′i, i = 1, 2)=(12)4N−µ(ω,k)−µ(ω′,k′)−µ(v,`). (16.24)Proof. The proofs of the height relations may be reproduced verbatim from theprevious lemmas in this chapter, so we focus only on the probability estimate. Asbefore,uN = DN(t1, t2) ∼= Qu[ω, k], u′N = DN(t′1, t′2) ∼= Qu′ [ω′, k′]zN = DN(uN, u′N) ∼= Qz[v, `] = Qu[w, `] = Qu′ [ω′, `], and hence (16.25)hN(uN) = µ(ω, k), hN(u′N) = µ(ω′, k′), hN(zN) = µ(v, `).Since ` ≤ k ≤ k′, (16.25) impliesuN ∪ u′N ⊆ zN, and thus µ(v, `) ≤ min[µ(ω, k), µ(ω′, k′)].It is important to keep in mind that N(A;α) need not inherit the same type ofstructure as A. For example, if (16.20) holds, it need not be true that uN ∩ u′N = ∅;indeed the vertices uN, u′N and zN could be distinct or (partially) coincident dependingon the structure of the slope tree. Nonetheless the information collected above issufficient to compute the number of vertices in N(A;α) (see Figure 16.3):n(A;α) = µ(v, `)︸ ︷︷ ︸vertices above zN+[µ(ω, k) − µ(v, `)]︸ ︷︷ ︸vertices between zN and uN+[µ(ω′, k′) − µ(v, `)]︸ ︷︷ ︸vertices between zN and u′N+ 2[N − µ(ω, k)]︸ ︷︷ ︸ancestors of t1 and t2in N descended from uN+ 2[N − µ(ω′, k′)]︸ ︷︷ ︸ancestors of t′1 and t′2in N descended from u′N= 4N − µ(ω, k) − µ(ω′, k′) − µ(v, `).Combined with Lemma 16.2, this leads to (16.24).Lemma 16.9. Let A = {ti, t′i; i = 1, 2} be a collection of four distinct root cubes176such that I = {(t1, t2); (t′1, t′2)} obeys (16.19) and is of type 2. Suppose that ΓA ={α(ti) = vi, α(t′i) = v′i ; i = 1, 2} is a choice of slopes such that the collection{(ti, vi); (t′i, v′i); i = 1, 2} is sticky-admissible. Setω = D(v1, v2), ω′ = D(v′1, v′2), k = h(u), k′ = h(u′),so that k < k′. Then the following inequalities hold: k ≤ λ(ω), k′ ≤ λ(ω′). Further,there exist permutations {i1, i2} and {j1, j2} of {1, 2} for which the quantitiesϑ = D(vi2 , v′j2), t = D(ti2 , t′j2), ` = h(t)obey the relation ` ≤ λ(ϑ), and for which the probability of slope assignment can becomputed as follows:Pr(σ(ti) = vi, σ(t′i) = v′i for i = 1, 2)=(12)4N−µ(ω,k)−µ(ω′,k′)−µ(ϑ,`). (16.26)Proof. The definition of the configuration type dictates that u′ is strictly containedin u, but depending on other properties of the ray joining u and u′ we are led toconsider several cases. If there does not exist any vertex in the root tree that is strictlycontained in u and also contains ti for some i = 1, 2, then any permutation of theroot pairs {t1, t2} and {t′1, t′2} works. In particular, it suffices to choose i1 = j1 = 1,i2 = j2 = 2. In this case t = u, hence ` = k. In particular this impliesθ(ω, k) = θ(v2, k) = θ(v2, `) = θ(ϑ, `), hence µ(ω, k) = µ(ϑ, `). (16.27)Furtheru′N = Qu′ [ω′, k′] = Qt′1 [v′1, k′] ⊆ Qt′1 [v′1, k] = Qu[ω, k] = uN, andhN(uN) = µ(ω, k), hN(u′N) = µ(ω′, k′).177Referring to Figure 16.3 we find thatn(A;α) = µ(ω, k) + 2[N − µ(ω, k)]+[µ(ω′, k′) − µ(ω, k)]+ 2[N − µ(ω′, k′)]= 4N − 2µ(ω, k) − µ(ω′, k′)= 4N − µ(ω, k) − µ(ω′, k′) − µ(ϑ, `),where the last step uses one of the equalities in (16.27).Suppose next that the previous case does not hold, and also that none of thedescendants of u′ lying on the rays of t′1, t′2 is an ancestor of t1 or t2. Then there is avertex, let us call it t, such that u′ ⊆ t ( u, and t is of maximal height in this classsubject to the restriction that it is an ancestor of some ti, which we call ti2 . Thusti1 is the unique element in {t1, t2} that is not a descendant of t. In this case, anypermutation of {t′1, t′2} works, and we can keep j1 = 1, j2 = 2. Then t = D(ti2 , t′j2),k < ` ≤ k′, andu′N = DN(t′1, t′2) ∼= Qu′ [ω′, k′] = Qu′ [v′j2 , k′],tN = DN(ti2 , t′j2) ∼= Qt[ϑ, `] = Qu′ [v′j2 , `], anduN = DN(t1, t2) ∼= Qu[ω, k] = Qu[ω0, k] = Qu′ [v′j2 , k],where the last line uses the fact that u = D(t1, t2, t′1, t′2), so that the second equalityin that line holds ω0 = D(v1, v2, v′1, v′2). These relations imply thatu′N ⊆ tN ⊆ uN with hN(uN) = µ(ω, k), hN(u′N) = µ(ω′, k′), hN(tN) = µ(ϑ, `). (16.28)Using this, we compute n(A;α) as follows,n(A;α) = µ(ω, k) +[N − µ(ω, k)]︸ ︷︷ ︸vertices of ti1 in N+[µ(ϑ, `) − µ(ω, k)]+[N − µ(ϑ, `)]︸ ︷︷ ︸vertices of ti2 in N below uN+[µ(ω′, k′) − µ(ϑ, `)]︸ ︷︷ ︸vertices between tN and u′N+ 2[N − µ(ω′, k′)]︸ ︷︷ ︸vertices of t′1 and t′2in N below u′N178= 4N − µ(ω, k) − µ(ω′, k′) − µ(ϑ, `),which is the required exponent.The last case, complementary to the ones already considered is when there existsa pair of indices, denoted i2, j2 ∈ {1, 2} such that t = D(ti2 , t′j2) ( u′. In this casewe leave the reader to verify by the usual means thattN ⊆ u′N ⊆ uN,with their heights given by the same expressions as in (16.28). Accordingly,n(A;α) = N︸︷︷︸vertices of ti1in N+[N − µ(ω, k)]︸ ︷︷ ︸vertices on t′j1in N below uN+[µ(ϑ, `) − µ(ω′, k′)]︸ ︷︷ ︸vertices between tN and u′N+ 2[N − µ(ϑ, `)]︸ ︷︷ ︸vertices of ti2 and t′j2in N below tN= 4N − µ(ω, k) − µ(ω′, k′) − µ(ϑ, `).Thus, despite structural differences, all the cases give rise to the same value of n(A;α)that agrees with the exponent in (16.26), completing the proof.We pause for a moment to record a few properties of the youngest commonancestors of the roots and slopes that emerged in the proof of Lemma 16.9.Corollary 16.10. Let A and ΓA be as in Lemma 16.9.(i) The possibly distinct vertices u, u′ and t, as described in Lemma 16.9, arelinearly ordered in terms of ancestry, i.e., there is some ray of the root tree thatthey all lie on. Depending on A, the vertex t may lie above or below u′, butalways in u.(ii) The splitting vertices ω, ω′, ϑ in the slope tree also obey certain inclusions;namely, for each of the pairs (ω, ϑ) and (ω′, ϑ), one member of the pair iscontained in the other.Proof. Both u′ and t lie on the ray identifying t′j2 by definition, and u lies on the rayof u′ by the assumption on the type of the root configuration. This establishes the179first claim. The definitions also imply that vi2 ⊆ ω ∩ ϑ and v′j2 ⊆ ω′ ∩ ϑ, hence bothintersections are non-empty. The second conclusion then follows from the nestingproperty of M -adic cubes.Lemma 16.11. Let A = {ti, t′i; i = 1, 2} be a collection of four distinct root cubessuch that I = {(t1, t2); (t′1, t′2)} is of type 3. Suppose that ΓA = {α(ti) = vi, α(t′i) =v′i ; i = 1, 2} is a choice of slopes such that the collection {(ti, vi); (ti, v′i); i = 1, 2} issticky-admissible. Setω = D(v1, v2), ω′ = D(v′1, v′2), k = h(u) = h(u′).Then the following relations must hold: k ≤ λ(ω), k ≤ λ(ω′), µ(ω, k) = µ(ω′, k).Further, there exist permutations {i1, i2} and {j1, j2} of {1, 2} such that the quantitiess1 = D(ti1 , t′j1), s2 = D(ti2 , t′j2), `1 = h(s1),ϑ1 = D(vi1 , v′j1), ϑ2 = D(vi2 , v′j2), `2 = h(s2)satisfys1 ⊆ u, s2 ( u, k ≤ `1 ≤ `2, `i ≤ λ(ϑi) for i = 1, 2, (16.29)and for whichPr(σ(ti) = vi, σ(t′i) = v′i for i = 1, 2)=(12)4N−µ(ω,k)−µ(ϑ1,`1)−µ(ϑ2,`2). (16.30)Proof. Since I is of type 3, u = u′ is the youngest common ancestor of the fourelements in I. If ω0 is the youngest common ancestor of the slopes {vi, v′i : i = 1, 2},then λ(ω0) ≥ h(u) = k by sticky admissibility. Thus θ(ω, k) = θ(ω0, k) = θ(ω′, k),and therefore µ(ω, k) = µ(ω′, k), as claimed.We turn to (16.29) and the probability estimate. The configuration type dictatesthat there exist indices (i, j) ∈ {1, 2}2 such that D(ti, t′j) ( u. Among all such pairs(i, j), we pick one for which D(ti, t′j) is of maximal height. Let us call this pair (i2, j2),180so that h(D(ti2 , t′j2)) ≥ h(D(ti, t′j)) for all 1 ≤ i, j ≤ 2. The first three relations in(16.29) are now immediate. The last one follows from sticky admissibility and is leftto the reader.It remains to compute n(A;α). The structure of N(A;α) gives thatuN = u′N = DN(t1, t2) = DN(t′1, t′2)uN = Qu[ω, k] = Qu′ [ω′, k] = Qt1 [v1, k] = Qt2 [v2, k],siN = DN(ti, t′i) = Qsi [ϑi, `i] = Qti [vi, `i],siN ⊆ uN = DN(s1N, s2N) for i = 1, 2, so thathN(uN) = µ(ω, k) ≤ hN(siN) = µ(ϑi, `i), i = 1, 2.Putting these together, the number of vertices in N(A;α) is obtained as follows,n(A;α) = µ(ω, k)︸ ︷︷ ︸vertices up to uN+2∑i=1[µ(ϑi, `i) − µ(ω, k)]︸ ︷︷ ︸vertices between uN and siN+2∑i=12[N − µ(ϑi, `i)]︸ ︷︷ ︸vertices below siN= 4N − µ(ϑ1, `1) − µ(ϑ2, `2) − µ(ω, k).The probability estimate claimed in (16.30) now follows from Lemma 16.2.Corollary 16.12. Let ω, ω′, ϑ1, ϑ2 be as in Lemma 16.9. Then each of the pairs(ω, ϑ1), (ω, ϑ2), (ω′, ϑ1) and (ω′, ϑ2) has the property that one member of the pair iscontained in the other.Proof. Since vi1 ⊆ ω ∩ ϑ1, vi2 ⊆ ω ∩ ϑ2, v′j1 ⊆ ω′ ∩ ϑ1 and v′j2 ⊆ ω′ ∩ ϑ2, allfour intersections are nonempty, and the desired conclusion follows from the nestingproperty of M -adic cubes.As the reader has noticed, the classification of probability estimates in this sectionis predicated on the configuration types of the roots, not the slopes. Of course, suchdefinitions of type apply equally well to slope tuples {(v1, v2); (v′1, v′2)}. Indeed, apoint worth noting is that configuration types are not preserved under even stickymaps; see for example the diagram in Figure 16.4 below, where a four tuple of roots181of type 1 maps to a sticky image of type 3. In view of these considerations, weshall refrain for the most part from using any type properties of slopes. In the rareinstances where structural properties of slopes are relevant, a case in point beingSection 19.2.3, we need to consider all possible configurations.D(t1, t2) D(t′1, t′2)t2t1 t′1 t′2 σ(t2)σ(t′1) σ(t1)σ(t′2)σFigure 16.4: An example of a four tuple of roots of type 1 mapping to a sticky imageof type 3. Notice that D(σ(t1), σ(t2)) = D(σ(t′1), σ(t′2)).182Chapter 17Tube countsA question of considerable import, the full significance of which will emerge inChapter 19, is the following: what is the maximum possible cardinality of a sticky-admissible collection of tube tuples that admit certain pairwise intersections in apre-fixed segment of space? The answer depends, among other things, on the sizeand configuration type of the roots of the tubes. In this section, we discuss thesesize counts for collections that are simple enough in the sense that an element inthe collection is either a pair, a triple or at most a quadruple of tubes, so that theconfiguration type of the roots has to fall in one of the categories described in Section16.2.17.1 Collections of two intersecting tubesLet us start with the case where the collection consists of pairs of tubes. To phrasethe question above in more refined terms we define a collection of root-slope tuplesE2[u, ω; %], where u is a vertex of the root tree, ω is a splitting vertex of the slope tree,and % ∈ [M−J , 10A0] is a constant that represents the (horizontal) distance from the183root hyperplane to where the intersection takes place.E2[u, ω; %] :={(t1, v1), (t2, v2)}sticky-admissible∣∣∣∣∣t1, t2 ∈ Q(J), u = D(t1, t2), t1 6= t2,v1, v2 ∈ ΩN , ω = D(v1, v2),Pt1,v1 ∩ Pt2,v2 ∩ [%, C1%] × Rd 6= ∅. (17.1)In this context the question at the beginning of this section can be restated as: what isthe cardinality of E2[u, ω; %]? We answer this question in Lemma 17.3 of this chapter,splitting the necessary work between two intermediate lemmas whose content willalso be used in later counting arguments. To be specific, Lemma 17.1 obtains auniform bound on a t2-slice of E2[u, ω; %] for fixed t1, v1 and v2. The cardinality ofthe projection of E2[u, ω; %] onto the t1 coordinate is obtained in Lemma 17.2.Lemma 17.1. Let E2[u, ω; %] be the collection defined in (17.1), and let ρω = sup{|a−b| : a, b ∈ ΩN , D(a, b) = ω} be the quantity defined in (12.7).(i) If E2[u, ω; %] is nonempty, then 2C1%ρω ≥M−J .(ii) Given a constant C1 > 0 used to define E2[u, ω; %], there exists a constantC2 = C2(d,M,C0, A0, C1) > 0 with the following property. For any fixed choiceof t1 ∈ Q(J) and v1, v2 ∈ ΩN the following estimate holds:#{t2 ∈ Q(J) : {(t1, v1), (t2, v2)} ∈ E2[u, ω; %]}≤ C2%ρωMJ . (17.2)Proof. The proof is illustrated in Figure 17.1. If {(t1, v1), (t2, v2)} is a tuple thatlies in E2[u, ω; %], then there exists x ∈ [%, C1%] × Rd such that x also belongs toPt1,v1 ∩ Pt2,v2 . By Lemma 7.1, an appropriate version of inequality (7.1) must hold,i.e., there exists x1 ∈ [%, C1%] such that|cen(t2) − cen(t1) + x1(v2 − v1)| ≤ 2cd√dM−J . (17.3)In conjunction with Corollary 7.2, this leads to the inequalityM−J ≤ |cen(t2) − cen(t1)| ≤ |x1||v2 − v1| + 2cd√dM−J184≤ 2|x1||v2 − v1| ≤ 2C1%ρω,which is the conclusion of part (i). The inequality (17.3) also implies that cen(t2) isconstrained to lie in a O(M−J) neighborhood of the line segmentcen(t1) − s(v2 − v1), % ≤ s ≤ C1%. (17.4)The length of this segment is at most C1%|v2 − v1| ≤ C1%ρω, since v1 and v2 mustlie in distinct children on ω. In view of part (i), the number of possible choices forM−J -separated points cen(t2), and hence for t2, lying within this neighborhood isO(%ρωMJ), as claimed in part (ii).x1Rdcen(t1)% C1%v1v2possiblecen(t2)Figure 17.1: An illustration of the proof of Lemma 17.1.Lemma 17.2. Given C1 > 0, there exists a positive constant C2 = C2(d,M,A0, C1)with the following property. For any E2[u, ω; %] defined as in (17.1), the following185estimate holds:#{t1 ∈ Q(J)∣∣∣∃t2 ∈ Q(J) and v1, v2 ∈ ΩN suchthat {(t1, v1); (t2, v2)} ∈ E2[u, ω; %]}≤ C2%ρωM−(d−1)h(u)+dJ ,(17.5)where ρω is as in (12.7).Proof. The proof is illustrated in Figure 17.2. If {(t1, v1), (t2, v2)} ∈ E2[u, ω; %], thenthere exists x = (x1, · · · , xd+1) ∈ Pt1,v1 ∩ Pt2,v2 with % ≤ x1 ≤ C1%. Combininginequality (17.3) obtained from Lemma 7.1 along with Corollary 7.2 as we did inLemma 17.1, we obtain|cen(t2) − cen(t1)| ≤ |x1||v2 − v1| + 2cd√dM−J≤ (1 + 4cd√d)|x1||v1 − v2|≤ C1(1 + 4cd√d)%ρω = C%ρω,(17.6)where the last step follows from the definition of ω. This means that cen(t1) andcen(t2) must be within distance C%ρω of each other. On the other hand, it is knownas part of the definition of E2[u, ω; %] that u = D(t1, t2), so cen(t1) and cen(t2) mustlie in distinct children of u. This forces the location of cen(t1) to be within distanceC%ρω of the boundary of some child of u, to allow for the existence of a point cen(t2)contained in a different child and obeying the constraint of (17.6). In other words,cen(t1) belongs to the setAu ={s ∈ u : dist(s, bdry(u′)) ≤ C%ρω for some child u′ of u}, (17.7)which is the union of at most dM parallelepipeds of dimension d, with length M−h(u)in (d − 1) “long” directions and C%ρω in the remaining “short” direction. Notethat ρω ≤ M−h(ω) ≤ M−h(u) by sticky-admissibility, hence %ρω = O(M−h(u)), whichjustifies this description.Since C%ρω ≥ M−J by Lemma 17.1(i), the constituent parallelpipeds of Au asdescribed above are thick relative to the finest scale M−J in all directions. The186volume of Au is then easily computed as|Au| ≤ C%ρωM−(d−1)h(u).Therefore the number of M−J separated points cen(t1), and hence the number ofpossible root cubes t1, contained in Au is at most C2%ρωM−(d−1)h(u)+dJ , as claimed.M−h(u)C%ρωFigure 17.2: Proof of Lemma 17.2 illustrated, with d = 2 and M = 3. The outermostsquare is u, and the smallest squares depict the root cubes in Au.Lemma 17.3. Let E2[u, ω; %] be the collection of pairs of tubes defined in (17.1).Then#(E2[u, ω; %]) ≤ C(%ρω)222(N−ν(ω))M−(d−1)h(u)+(d+1)J .Here ν(ω) denotes the index of the splitting vertex ω, as defined in (12.4).187Proof. We combine the counts from Lemmas 17.1 and 17.2. For fixed t1, v1 and v2,the number of possible t2 such that {(t1, v1), (t2, v2)} ∈ E2[u, ω; %] is bounded aboveby the quantity on the right hand side of (17.2). The number of possible t1 is at mostthe right hand side of (17.5), whereas the number of possible v1, hence also v2, is2N−ν(ω) due to the binary nature of ΩN as discussed in Section 12.3. The claimed sizeestimate of E2[u, ω; %] is simply the product of all the quantities mentioned above.17.2 Counting slope tuplesVariations of the arguments presented in Section 17.1 also apply to more generalcollections. For the proof of the lower bound (14.9), we will need to estimate, inaddition to the above, the sizes of collections consisting of tube triples and tubequadruples with certain pairwise intersections. The collections of tube tuples whosecardinalities are of interest are analogues of E2[u, ω; %] of greater complexity, and theirconstructions share the common feature that the probability of slope assignment forany tube tuple within a collection is constant and falls into one of the categoriesclassified in Chapter 16. As we have seen in that chapter, the probability depends,among other things, on certain splitting vertices of the slope tree occurring as pair-wise youngest common ancestors. In particular, which subset of pairwise youngestcommon ancestors has to be considered, whether for root or slope, is dictated by theroot configuration type. An important component of tube-counting is therefore toestimate how many possible slope tuples can be generated from a given set of suchsplitting vertices. Before moving on to the main counting arguments in this chapterpresented in Sections 17.3 and 17.4, we observe a few facts that help in countingtuples of slopes, given some information about their ancestry.Lemma 17.4. (i) Given any Γ ⊆ ΩN , #(Γ) ≤ 4, there exist at most three distinctvertices {$i : i = 1, 2, 3} ⊆ G(ΩN) with the propertiesh($1) ≤ h($2) ≤ h($3), $2, $3 ⊆ $1, (17.8)such that D(w,w′) ∈ {$i : i = 1, 2, 3} for any w 6= w′, w,w′ ∈ Γ. Of course,188the containment relations in (17.8) imply that λ($2), λ($3) ≤ λ($1).(ii) Suppose now that we are given {$i : i = 1, 2, 3}, possibly distinct splittingvertices of the slope tree obeying (17.8). Definem = m[$1, $2, $3] :=2(ν($3) + ν($2)) if $3 6⊆ $2,2ν($3) + ν($2) + ν($1) if $3 ⊆ $2.(17.9)Fix three distinct pairs of indices {(ik, jk) : ik 6= jk, 1 ≤ k ≤ 3} ⊆ {1, 2, 3, 4}2with the property that⋃{ik, jk : k = 1, 2, 3} = {1, 2, 3, 4}. Then#{(w1, w2, w3, w4) ∈ Ω4N : D(wik , wjk) = $k, 1 ≤ k ≤ 3}≤ C24N−m.Proof. If Γ is given, we arrange all the pairwise youngest common ancestors of Γ, i.e,the vertices in DΓ := {D(w,w′) : w 6= w′, w, w′ ∈ Γ}, in increasing order of height,where distinct vertices of the same height can be arranged in any way, say accordingto the lexicographic ordering. We define $3 to be a vertex of maximal height inDΓ, and $2 to be a vertex of maximal height in DΓ \ {$3}. Due to maximality ofheight and the binary nature of the slope tree as ensured by Proposition 12.1, $3has exactly two descendants in Γ, say w1 and w2.If $3 6⊆ $2, then there is no overlap among the descendants of these two vertices.Thus the two descendants w3 and w4 of $2 must be distinct from w1, w2, thusaccounting for all the elements of Γ. In this case the conclusion of the lemma holdswith $1 = D($2, $3). If $3 ( $2, then again by maximality of height $2 cancontribute exactly one member of Γ that is neither w1 nor w2. Let us call this newmember w3. If #(Γ) = 3, then the proof is completed by setting $1 = D($2, $3) =$2. If #(Γ) = 4, we call the remaining child w4, which is not descended from $2,and set $1 = D($2, w4). This selection meets (17.8), and also accounts for all thepairwise youngest common ancestors of Γ, as required by part (i) of the lemma.A very similar argument can be used to prove part (ii). Since the total number ofslopes in ΩN generated by $3 is exactly 2N−ν($3)+1, this is the maximum number ofpossible choices for each of wi3 and wj3 . If $3 6⊆ $2, then {i2, j2}∩{i3, j3} = ∅. Since189each of wi2 and wj2 admits at most 2N−ν($2)+1 possibilities by the same reasoning,the size of possible four tuples (w1, w2, w3, w4) in this case is at most 2 raised to thepower 2(N−ν($3)+1)+2(N−ν($2)+1), which gives the claimed estimate. If $3 ⊆$2 ⊆ $1, then by our assumptions on ik, jk, there exist indices `2 ∈ {i2, j2} \ {i3, j3}and `1 ∈ {i1, j1} \ {i3, j3, `2}. Since i3, j3, `1, `2 are distinct indices and the numberof possible choices of wi3 , wj3 , w`1 and w`2 are at most 2N−ν($3), 2N−ν($3), 2N−ν($1)and 2N−ν($2) respectively, the result follows.Minor modifications of the argument above yield the following analogue for slopetriples. The proof is left to the reader.Lemma 17.5. (i) Given a collection Γ ⊆ ΩN , #(Γ) ≤ 3, it is possible to rearrangethe collection of vertices {D(w,w′);w 6= w′, w, w′ ∈ Γ} as {$1, $2} with$2 ⊆ $1.(ii) Given a pair {$1, $2} ⊆ G(ΩN) with $2 ⊆ $1, definem̂ = m̂[$1, $2] := 2ν($2) + ν($1). (17.10)Let (i1, j1) 6= (i2, j2) be two pairs of indices such that {i1, j1, i2, j2} = {1, 2, 3}.Then the following estimate holds:#{(w1, w2, w3) : D(wi1 , wj1) = $1, D(wi2 , wj2) = $2} ≤ 23N−m̂.17.3 Collections of four tubes with at least twopairwise intersections17.3.1 Four roots of type 1We start with the simplest and generic situation, when the root quadruple is of type 1.Motivated by the expression of the probability obtained in (16.24), let us first fix twovertex triples (u, u′, z) and (ω, ω′, v) in the root tree and slope tree respectively thatsatisfy the height and containment relations prescribed in Lemma 16.8. For such a190selection and with % ∈ [M−J , 10A0], we define a collection E41 = E41[u, u′, z;ω, ω′, v; %]of sticky-admissible tube quadruples of the form {(t1, v1), (t2, v2), (t′1, v′1), (t′2, v′2)},obeying the following restrictions:I = {(t1, t2); (t′1, t′2)} is of type 1, t1 6= t2, t′1 6= t′2, u = D(t1, t2),u′ = D(t′1, t′2), z = D(u, u′), ω = D(v1, v2), ω′ = D(v′1, v′2), v = D(ω, ω′),Pt1,v1 ∩ Pt2,v2 ∩ [%, C1%] × Rd 6= ∅, Pt′1,v′1 ∩ Pt′2,v′2 ∩ [%, C1%] × Rd 6= ∅.(17.11)The result below provides a bound on the size of E41.Lemma 17.6. There exists a constant C > 0 such that#(E41)≤ C(%2ρωρω′)224N−2(ν(ω)+ν(ω′))M−(d−1)[h(u)+h(u′)]+2(d+1)J .Proof. Since the intersection and ancestry conditions imply thatE41[u, u′, z;ω, ω′, v] ⊆ E2[u, ω; %] × E2[u′, ω′; %],the stated size bound for E41 is the product of the sizes of the two factors on theright. These are obtained from Lemma 17.3 in Section 17.1, applied twice.17.3.2 Four roots of type 2The treatment of this case follows a similar route, though with certain importantvariations. The main distinction from Section 17.3.1 is that the intersection and typerequirements place greater constraints on the selection of the roots and slopes, andhence on the number of tube quadruples. Thus better bounds are possible, comparedto the trivial ones exploited in Lemma 17.3.Let (u, u′, t) and (ω, ω′, ϑ) be vertex triples in the root tree and slope tree respec-tively that meet the requirement of Corollary 16.10. In other words, the verticesu, u′, t are linearly ordered in terms of ancestry, and obey u′ ( u, while ω ∩ ϑ 6= ∅and ω′ ∩ ϑ 6= ∅. Holding these fixed, define E42 = E42[u, u′, t;ω, ω′, ϑ; %] to be the col-lection of all sticky-admissible tuples of the form {(ti, vi), (t′i, v′i) : i = 1, 2} obeying191the properties:I = {(t1, t2); (t′1, t′2)} is of type 2, u′ = D(t′1, t′2) ( u = D(t1, t2),t = D(t2, t′2), ω = D(v1, v2), ω′ = D(v′1, v′2), ϑ = D(v2, v′2),Pt1,v1 ∩ Pt2,v2 ∩ [%, C1%] × Rd 6= ∅, Pt′1,v′1 ∩ Pt′2,v′2 ∩ [%, C1%] × Rd 6= ∅.(17.12)The vertex triple (ω, ω′, ϑ) obeys the hypothesis of Lemma 17.4(ii), permitting theapplication of this lemma in the counting argument presented in Lemma 17.8.Lemma 17.7. If the vertex pairs (ω, ϑ) and (ω′, ϑ) both have the property that onemember of the pair is contained in the other, then there exists a rearrangement of{ω, ω′, ϑ} as {$1, $2, $3} that meets the requirement (17.8).Proof. If ω ∩ ω′ = ∅, then ϑ must contain both ω and ω′. In this case, we renameϑ as $1 and call $3 the element of {ω, ω′} with greater height. If ω ∩ ω′ 6= ∅, thenthe inclusion requirements imply that there must be a ray which contains all threevertices. Since the vertices are linearly ordered, we rename them based on height.Lemma 17.7 above allows us to define the quantity m as in (17.9), which by aslight abuse of notation we denote by m[ω, ω′, ϑ]. We are now in a position to statethe main result of this subsection, namely the size estimate for E42. The location oft relative to u, u′ affects the size estimate of E42, even though we have seen that theprobability estimate in (16.26) remains unchanged with respect to this property.Lemma 17.8. The following conclusions hold:(i) If u′ ⊆ t ⊆ u, then E42 is non-empty only if dist(t, bdry(u∗)) ≤ C%ρω. Here u∗is defined to be the unique child of u containing t if t ( u and is set to be equalto u if t = u. In either case,#(E42) ≤ C(%3ρ2ω′ρω)min[%ρω,M−h(t)]24N−m[ω,ω′,ϑ]×M−(d−1)(h(t)+h(u′))+2(d+1)J .192(ii) If t ( u′ ( u, then E42 is non-empty only ifdist(t, bdry(u∗)) ≤ C%ρω and dist(t, bdry(u′∗)) ≤ C%ρω′ ,where u∗, u′∗ are the children of u, u′ respectively that contain t. In this case,#(E42)≤ C(%2ρωρω′)24N−m[ω,ω′,ϑ] min[%ρω,M−h(t)]× min[%ρω′ ,M−h(t)]M−2(d−1)h(t)+2(d+1)J .Proof. Both statements in the lemma involve similar arguments. We only prove part(i) in detail, and leave a brief sketch for the other part. The argument here followsthe basic structure of Lemma 17.3, since we still have the trivial containmentE42[u, u′, t;ω, ω′, ϑ; %] ⊆ E2[u, ω; %] × E2[u′, ω′; %], (17.13)but with a few modifications resulting from the more refined information aboutthe roots and slopes available from t and ϑ. For instance, combining the definingassumptions that t2 ⊆ t and u = D(t1, t2) with the intersection inequality |cen(t2)−cen(t1)| ≤ 2C1%ρω derived from (17.3) in Lemma 17.1, we deduce that t has to liewithin distance 2C1%ρω of the boundary of u∗. This is the first conclusion of part(i). For the size bound, we reason as follows. By Lemma 17.1(ii), the number of t1and t′1, if everything else is held fixed, is ≤ C(%ρωMJ)(%ρω′MJ) ≤ C%2ρωρω′M2J .Turning to slope counts, we apply Lemma 17.4(ii), the use of which has already beenjustified in Lemma 17.7, to deduce that the number of possible slope quadruples(v1, v2, v′1, v′2) is 24N−m. It remains to compute the size of the t2 and t′2 projections ofE42. In view of (17.13), a bound on the size of the t′2 projection is given by the righthand side of (17.5) with u replaced by u′. On the other hand, t2 is restricted to liewithin t and within distance 2C1%ρω from the boundary of t if t ( u. This places anontrivial spatial restriction on t2 only if 2C1%ρω < M−h(t). If t = u, the argumentleading up to (17.5) shows that t2 lies in Au defined in (17.7). In either event thevolume of the region where t2 can range is at most C min(%ρω,M−h(t))M−(d−1)h(t),hence the cardinality of the t2 projection is at most MdJ times this quantity (see193Figure 17.3). Combining all these counts yields the bound on the size of E42 givenin part (i).utu′M−h(t)2C1%ρωFigure 17.3: Illustration of the spatial restriction on t2 imposed by the conditionsu′ ⊂ t ⊂ u, t2 ⊂ t, dist(t1, t2) ≤ 2C1%ρω < M−h(t). Here, t2 must lie within theshaded region along the boundary of t, with t1 falling just outside this boundary inthe unshaded thatched region.For part (ii), the size estimate of E42 is a product of a number of factors analogousto the ones already considered, the origins of which are indicated below.#(t1 given v1, v2, t2) ≤ C%ρωMJ ,#(t′1 given v′1, v′2, t′2) ≤ C%ρω′MJ ,}(Lemma 17.1(ii))#(t2) ≤ C min[%ρω,M−h(t)]M−(d−1)h(t)+dJ ,#(t′2) ≤ C min[%ρω′ ,M−h(t)]M−(d−1)h(t)+dJ ,}(arguments similar to part (i)),#(v1, v2, v′1, v′2) ≤ 24N−m[ω,ω′,ϑ] (from Lemma 17.4(ii)).We omit the details.17.3.3 Four roots of type 3To complete the discussion of size for collections consisting of intersecting tubequadruples, it remains to consider the case where the root configuration is of type 3.194Motivated by the conclusions of Lemma 16.11 and Corollary 16.12, we fix two vertextuples (u, s1, s2) and (ω, ω′, ϑ1, ϑ2) in the root tree and the slope tree respectively,with the properties that s1, s2 ⊆ u, h(u) ≤ h(s1) ≤ h(s2), ω∩ϑi 6= ∅, and ω′∩ϑi 6= ∅for i = 1, 2. For such a selection, we define E43[u, s1, s2;ω, ω′, ϑ1, ϑ2; %] to be thecollection of all sticky-admissible tuples {(ti, vi), (t′i, v′i) : i = 1, 2} that satisfy the listof conditions below:I = {(t1, t2); (t′1, t′2)} is of type 3, u = D(t1, t2) = D(t′1, t′2),ω′ = D(v′1, v′2) ⊆ ω = D(v1, v2), si = D(ti, t′i), ϑi = D(vi, v′i), i = 1, 2,Pt1,v1 ∩ Pt2,v2 ∩ [%, C1%] × Rd 6= ∅, Pt′1,v′1 ∩ Pt′2,v′2 ∩ [%, C1%] × Rd 6= ∅.(17.14)Since I is of type 3, interchanging (t1, t2) and (t′1, t′2) leaves u unchanged. Hence wemay assume without loss of generality that ρω ≤ ρω′ . Further, Lemma 17.4(i) dictatesthat for E43 to be non-empty, at most three out of the four vertices ω, ω′, ϑ1, ϑ2 canbe distinct. We leave the reader to verify that Lemma 17.7 can be applied to anytriple of these four vertices. Thus for any choice of an eligible tuple {ω, ω′, ϑ1, ϑ2},there exists a rearrangement of its entries as {$1, $2, $3} obeying the hypothesisand hence the conclusion of Lemma 17.4(ii). This permits a consistent definition ofthe quantity m[ω, ω′, ϑ1, ϑ2] as in (17.9), which we use in the statement of the lemmabelow. (Note that #({ω, ω′, ϑ1, ϑ2}) ≤ 3.)Lemma 17.9. If si ( u, let ui denote the child of u that contains si. Set ∆ :=min[%ρω, %ρω′ ].(i) The collection E43 is nonempty only if2∑i=1dist(si, bdry(ui))≤ C∆, (17.15)where dist(s1, bdry(u1)) is defined to be zero if u = s1.(ii) If ∆ ≤ M−h(s1) and E43 is nonempty, then in addition to (17.15), one of thefollowing two conditions must hold:1951. s2 ( s1 = u, in which case s2 lies within distance C∆ of the boundary ofsome child of s1 = u.2. s2 ∩ s1 = ∅, in which case dist(s2, bdry(s1)) ≤ C∆.In either case, s2 is constrained to lie in the union of at most 2dM slab-likeparallepipeds, each with (d− 1) “long” directions of sidelength M−h(s1) and one“short” direction of sidelength ∆.(iii) If ∆ ≥ M−h(s1) and E43 is nonempty, then in addition to (17.15), s2 has tolie within a thin tube-like parallelepiped of length %min(M−h(ω),M−h(ω′)) inone “long” direction and thickness CM−h(s1) in the remaining (d− 1) “short”directions; more precisely, both the following inequalities must hold:|cen(s2) − cen(s1) + x1(cen(ω ∩ ϑ2) − cen(ω ∩ ϑ1))| ≤ CM−h(s1), and(17.16)|cen(s2) − cen(s1) + x′1(cen(ω′ ∩ ϑ2) − cen(ω′ ∩ ϑ1))| ≤ CM−h(s1) (17.17)for some x1, x′1 ∈ [%, C1%]. Here cen(t) denotes the centre of the cube t.(iv) In all cases, if E43 is nonempty,#(E43) ≤ C24N−m[ω,ω′,ϑ1,ϑ2]M−2(d−1)h(s2)+2(d+1)J×2∏i=1[min[%ρω,M−h(si)]min[%ρω′ ,M−h(si)]].Proof. Let us fix a tuple {(ti, vi); (t′i, v′i) : i = 1, 2} in E43. As in previous proofs suchas Lemma 17.2 (applications of which have appeared in the counting arguments ofLemma 17.3 and 17.8), the key elements are the inequalities|cen(t2) − cen(t1)| ≤ C%ρω and |cen(t′2) − cen(t′1)| ≤ C%ρω′ . (17.18)They are proved exactly in the same way as (17.6) follows from (17.3), resultingfrom the nontrivial intersection conditions that define E43. Combined with the set196inclusion relations u = D(t1, t2) = D(t′1, t′2) and t1, t′1 ⊆ s1 and t2, t′2 ⊆ s2 that areguaranteed by the type assumption on the roots, this yields thatdist(si, bdry(ui)) ≤ inf[dist(ti, bdry(ui)), dist(t′i, bdry(ui))]= inf[dist(ti, uci), dist(t′i, uci)]≤ inf[dist(t1, t2), dist(t′1, t′2)]≤ C min[%ρω, %ρω′ ] = C∆,leading to the distance constraints in (17.15). Incidentally, the inequalities (17.18)also prove the relation in part (ii) if s2 ∩ s1 = ∅. On the other hand, if s2 ⊆ s1, thens1 = u and the desired inequality is simply a restatement of the one in (17.15). Forpart (iii), we refer again to the intersection inequality (17.3), using it to deduce that∣∣cen(s2) − cen(s1) + x1(cen(ω ∩ ϑ2) − cen(ω ∩ ϑ1))∣∣≤2∑i=1|cen(si) − cen(ti)| + |x1|2∑i=1∣∣cen(ω ∩ ϑi) − vi∣∣+ |cen(t2) − cen(t1) + x1(v2 − v1)|≤√d2∑i=1M−h(si) + C1%√d2∑i=1M−h(ϑi) + 2cd√dM−J ≤ CM−h(s1).Here we have also used the height and inclusion relations associated with the rootconfiguration type established in Lemma 16.11; namely,ti ⊆ si, vi ∈ ω ∩ ϑi, h(si) ≤ h(ϑi), h(s1) ≤ h(s2), i = 1, 2.The inequality above implies that s2 has to lie within distance O(M−h(s1)) of a linesegment of length at most %|cen(ω ∩ ϑ2) − cen(ω ∩ ϑ1)| ≤ %M−h(ω). The inequality(17.17) is proved in an identical manner, using t′i, v′i, ω′ instead of ti, vi, ω. The firststatement in part (iii) is a consequence of both these inequalities.The bound on the size of E43 uses the same machinery as in the proof of Lemma17.8, so we simply indicate the breakdown of the contributions from the different197sources:#(E43) ≤ C min[%ρω,M−h(s1)]MJ︸ ︷︷ ︸#(t1) with t2, v1, v2 fixed× C min[%ρω′ ,M−h(s1)]MJ︸ ︷︷ ︸#(t′1) with t′2, v′1, v′2 fixed× 24N−m[ω,ω′,ϑ1,ϑ2]︸ ︷︷ ︸#(v1, v2, v′1, v′2)from Lemma 17.4(ii)× C min[%ρω,M−h(s2)]M−(d−1)h(s2)+dJ︸ ︷︷ ︸#(t2-projection)× min[%ρω′ ,M−h(s2)]M−(d−1)h(s2)+dJ︸ ︷︷ ︸#(t′2-projection),which leads to the stated estimate.17.4 Collections of three tubes with at least twopairwise intersectionsFor the sake of completeness and book-keeping, we record in this section the cardi-nality of collections consisting of intersecting tube triples. No new ideas are involvedin the proofs, which are in fact simpler than the ones in Section 17.3. These are leftto the interested reader.Using the notation set up in Lemmas 16.5 and 16.6, we define the collections E31 =E31[u, u′;ω, ω′; %] and E32 = E32[u, t;ω, ω′, ϑ; %] in exactly the same way E4i were de-fined. Namely, E3i consists of all sticky-admissible tuples of the form {(t1, v1), (t2, v2), (t′2, v′2)}such that I = {t1, t2, t′2} is of type i andPt1,v1 ∩ Pt2,v2 ∩ [%, C1%] × Rd 6= ∅, Pt1,v1 ∩ Pt′2,v′2 ∩ [%, C1%] × Rd 6= ∅.In addition, the members of E3i must satisfyu = D(t1, t2), u′ = D(t1, t′2), ω = D(v1, v2), ω′ = D(v1, v′2),with u = u′, t = D(t2, t′2) and ϑ = D(v2, v′2) if i = 2. We also define the quantitiesm̂[ω, ω′] for E31 and m̂[ω, ω′, ϑ] for E32; both are expressed using the formula (17.10),where {$1, $2} with $2 ⊆ $1 is a rearrangement of {ω, ω′} for E31 and of {ω, ω′, ϑ}for E32, by virtue of Lemma 17.5. With this notation in place, the size estimates on198E3i are as follows.Lemma 17.10. (i) Set ∆ := min(%ρω, %ρω′). Then#(E31) ≤ C∆%2ρωρω′23N−m̂[ω,ω′]M−(d−1)(h(u)+h(u′))+(2d+1)J .(ii) With the same definition of ∆ as in part (i),#(E32) ≤ C∆ min[%ρω,M−h(t)] min[%ρω′ ,M−h(t)]23N−m̂[ω,ω′,ϑ]M−2(d−1)h(t)+(2d+1)J .199Chapter 18Sums over root and slope verticesA recurrent feature of the proof of (14.9), as we will soon see in Chapter 19, is theuse of certain sums over specific subsets of vertices in the root and slope trees. Werecord the outcomes of these summation procedures in this section for easy referencelater.Lemma 18.1. Fix a vertex $0 ∈ G(ΩN), i.e. $0 is a splitting vertex of the slopetree. Then the following estimates hold.(i) For any α ∈ R,∑$∈G(ΩN )$⊆$02−αν($) ≤Cα2−αν($0) if α > 1,N2−ν($0) if α = 1,Cα2−αν($0)+N(1−α) if α < 1.(ii) For M ≥ 2, β > 0 and α ≥ 1,∑$∈G(ΩN )$⊆$0M−βλ($)2−αν($) ≤ Cα,βM−βλ($0)2−αν($0).Proof. By Proposition 12.5, the number of splitting vertices descended from $0 with200the property that ν($) = ν($0) + j is 2j. Since j can be at most N , we see that∑$∈G(ΩN )$⊆$02−αν($) ≤∑j2−α(ν($0)+j)2j ≤ 2−αν($0)N∑j=12j(1−α),from which part (i) follows. On the other hand, if ν($) = ν($0) + j, then λ($) −λ($0) ≥ ν($) − ν($0) = j. Thus, a similar computation shows that∑$∈G(ΩN )$⊆$0M−βλ($)2−αν($) =∑jM−β(λ($0)+j)2−α(ν($0)+j)2j≤M−βλ($0)2−αν($0)∞∑j=1M−βj2−(α−1)j.The last sum in the displayed expression is convergent, establishing the desired con-clusion in part (ii).Lemma 18.2. Fix a vertex y in the root tree and a splitting vertex $ in the slopetree such that h(y) ≤ h($). Given a constant β, one of the following estimates holdsfors(β) :=′∑zM−βh(z)2µ($,h(z)),where the sum∑′ takes place over all vertices z of the root tree such that z ⊆ y andh(z) ≤ λ($).(i) If β < d, then s(β) ≤ Cβ2ν($)M (d−β)λ($)−dh(y).(ii) If β = d, then s(d) ≤ C2ν($)h($)M−dh(y).(iii) If β > d, then s(β) ≤ Cβ2ν($)M−βh(y).(iv) If β > d is large enough so that 2Md < Mβ, then s(β) ≤ CβM−dh(y).Proof. Since $ is a splitting vertex of the slope tree, there exists an integer j ∈ [1, N ]such that $ ∈ Gj(ΩN), i.e., ν($) = j. By definition, every jth splitting vertex is201either itself a (j − 1)th basic slope cube or is contained in one. Let $` ∈ H`(ΩN) bethe `th slope cube that contains $, so that$1 ) $2 ) · · · ) $j−1 ⊇ $.If z is a vertex of the root tree such that h($`−1) ≤ h(z) < h($`) for some ` ≤ j−1,then µ($, h(z)) = ` − 1; on the other hand, if h($j−1) ≤ h(z) ≤ λ($), thenµ($, h(z)) = j − 1. This suggests decomposing the sum defining s(β) according tothe heights of the slope cubes containing $. Implementing this and recalling that#{z : z ⊆ y, h(z) = k} = Mdk−dh(y), we obtains(β) =j−1∑`=1h($`)−1∑k=h($`−1)2`−1′∑z:h(z)=kM−βk +λ($)∑k=h($j−1)2j−1′∑z:h(z)=kM−βk≤ C[j∑`=1λ($)∑k=h(y)2`−1M−βkMdk−dh(y)]≤ CM−dh(y)j∑`=12`−1λ($)∑k=h(y)M (d−β)k≤ CM−dh(y)2jλ($)∑k=h(y)M (d−β)k≤ C2ν($)M−dh(y)M (d−β)λ($) if β < d,λ($) if β = d,M−(β−d)h(y) if β > d.Upon simplification, these are the estimates claimed in parts (i)-(iii) of the lemma.Part (iv) follows from the observation that µ($, h(z)) ≤ h(z), hences(β) ≤′∑zM−βh(z)2h(z) ≤∑k2kM−βk+d(k−h(y))202≤M−dh(y)∑k(2MdMβ)k≤ CβM−dh(y).In view of spatial constraints on the ancestors of root cubes as encountered inLemmas 17.8 and 17.9, occasionally the sums that we consider take place over morerestricted ranges of vertices than the one in Lemma 18.2, even though the summandsmay retain the same form. The next result makes this quantitatively precise. Let$ be a splitting vertex of the slope tree, and R a fixed parallelepiped in the roothyperplane with sidelength β in (d−r) directions and γ in the remaining r directions,where 1 ≤ r ≤ d − 1 and β ≥ γ ≥ M−J . Given constants  ≥ M−λ($) and α ∈ R,we defines± = s±(α, ,R, $) :=∑z∈Z±M−αh(z)2µ(ω,h(z)), (18.1)where the index sets Z± are collections of vertices of the root tree defined as follows:Z := {z ⊆ R : h(z) ≤ λ($), M−h(z) ≤ },Z+ := Z ∩ {z : M−h(z) ≥ γ},Z− := Z ∩ {z : M−h(z) ≤ γ}.Lemma 18.3. The following estimates hold for s± defined in (18.1).(i) If α > d− r and  ≥ γ then s+ ≤ C2ν($)βd−rα−d+r.(ii) If α > d, then s− ≤ C2ν($)βd−rγr min(, γ)α−d.Proof. We have already established in the proof of Lemma 18.2 that µ($, h(z)) ≤ν($) − 1. Further if M−k ≥ γ, then there can be at most a constant numberof possible choices of kth generation M -adic cubes z that are contained in R andintersect with a slice of R that fixes coordinates in the (d− r) long directions. Thuswe only need to count the number of possible z in the long directions, obtaining#{z ∈ Q(k) : z ⊆ R} ≤ Crβd−rM (d−r)k. (18.2)203Taking this into account, we obtains+ ≤ 2ν($)∑k:γ≤M−k≤M−αkβd−rM (d−r)k ≤ C2ν($)βd−rα−d+r,as claimed in part (i). Part (ii) follows in an identical manner; the only difference isthat now all directions of R are thick relative to the scale of z, hence (18.2) has tobe replaced by#{z ∈ Q(k) : z ⊆ R} ≤ Cγrβd−rMdk.204Chapter 19Proof of the lower bound (14.9)We are now in a position to complete the proof of Proposition 14.2 by verifying theprobabilistic statement on the lower bound of KN(X) claimed in (14.9). The twopropositions stated below are the main results of this chapter and allow passage tothis final step.Proposition 19.1. Fix integers N and R with N  M and 10 ≤ R ≤ 110 logM N .DefineP ∗t,σ,R := Pt,σ(t) ∩ [M−R,M−R+1] × Rd,where σ = σX is the randomized weakly sticky map described in Section 14. Thenthere exists a constant C = C(M,d) > 0 such thatEX[∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣]≤ CNM−2R. (19.1)Proposition 19.2. Under the same hypotheses as Proposition 19.1,EX[(∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣)2]≤ C(NM−2R)2. (19.2)Propositions 19.1 and 19.2 should be viewed as the direct generalizations ofPropositions 10.1 and 10.2 for arbitrary direction sets. These are proved belowin Sections 19.1 and 19.2 respectively. Of the two results, Proposition 19.2 is of205direct interest, since it leads to (14.9), as we will see momentarily in Corollary 19.3.Proposition 19.1, while not strictly speaking relevant to (14.9), nevertheless providesa context for presenting the core arguments within a simpler framework.Corollary 19.3. Proposition 19.2 implies (14.9).Proof. The argument here is identical to Corollary 10.3, and is omitted.19.1 Proof of Proposition 19.1Proof. We first recast the sum on the left hand side of (19.1) in a form that bringsinto focus its connections with the material in Chapters 16 and 17. By the Co´rdobaestimate, inequality (1.5),∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣ ≤∑1CdM−(d+1)J|σ(t1) − σ(t2)| +M−J, (19.3)where∑1 denotes the sum over all root pairs (t1, t2) such that t1 6= t2 and P ∗t1,σ,R ∩P ∗t2,σ,R 6= ∅. Unravelling the implications of the intersection we find that{(t1, t2) : t1 6= t2, P ∗t1,σ,R ∩ P∗t2,σ,R 6= ∅}⊆(t1, t2)∣∣∣∣∣∃ a unique pair (v1, v2) ∈ Ω2N such thatPt1,v1 ∩ Pt2,v2 ∩ [M−R,M−R+1] × Rd 6= ∅,and σ(t1) = v1, σ(t2) = v2, t1 6= t2. (19.4)For a given root pair (t1, t2), there may exist more than one slope pair (v1, v2) thatmeets the intersection criterion in (19.4). But only one pair will also satisfy, for agiven σ, the requirement σ(t1) = v1, σ(t2) = v2, which explains the uniqueness claimin (19.4). Using this, the expression on the right hand side of (19.3) can be expandedas follows,∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣ ≤∑1CdM−(d+1)J|σ(t1) − σ(t2)|206≤∑2CdM−(d+1)J|v1 − v2|T ((t1, v1), (t2, v2))≤∑u,ωCdM−(d+1)Jδω∑3T ((t1, v1), (t2, v2)), (19.5)where the notation∑2 in the second step denotes summation over the collection in(19.4), and T ((t1, v1), (t2, v2)) is a binary (random) counter given byT ((t1, v1), (t2, v2)) =1 if σ(t1) = v1 and σ(t2) = v2,0 otherwise.(19.6)In the last step (19.5) of the string of inequalities above, we have rearranged thesum in terms of the youngest common ancestors u = D(t1, t2) and ω = D(v1, v2)in the root tree and in the slope tree respectively. The summation∑3 takes placeover all sticky-admissible tube pairs {(t1, v1), (t2, v2)} in the deterministic collectionE2[u, ω; %] defined in (17.1), with % = %R = M−R and C1 = M . Incidentally, therequirement of sticky-admissibility restricts u and ω to obey the height relationh(u) < λ(ω). The quantity δω has been defined in (12.8), and is therefore ≤ |v1−v2|.With this preliminary simplification out of the way, we proceed to compute theexpected value of the expression in (19.5), combining the geometric facts and countingarguments from Chapter 17 with appropriate probability estimates from Chapter 16.Accordingly, we getEX[∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣]≤∑u,ωCdM−(d+1)Jδω∑E2[u,ω;%]EX[T ((t1, v1), (t2, v2))]≤∑u,ωCdM−(d+1)JC2ρω︸ ︷︷ ︸Corollary 12.4∑E2[u,ω;%]Pr(σ(t1) = v1, σ(t2) = v2)≤ CM−(d+1)J∑u,ωρ−1ω #(E2[u, ω; %])(12)2N−µ(ω,h(u))︸ ︷︷ ︸Lemma 16.3207≤ CM−(d+1)J∑u,ωρ−1ω (%ρω)222(N−ν(ω))M−(d−1)h(u)+(d+1)J︸ ︷︷ ︸Lemma 17.3×(12)2N−µ(ω,h(u))≤ CM−2R∑u,ωM−λ(ω)−(d−1)h(u)2µ(ω,h(u))−2ν(ω),where the last step uses the fact that ρω ≤ C1M−λ(ω). To establish the conclusionclaimed in (19.1), it remains to show that the last expression in the displayed stepsabove is bounded by CN . This follows from a judicious use of the summation resultsproved in Chapter 18; namely,∑u,ωM−λ(ω)−(d−1)h(u)2µ(ω,h(u))−2ν(ω) =∑ω∈GM−λ(ω)2−2ν(ω)∑uM−(d−1)h(u)2µ(ω,h(u))≤ C∑ω∈GM−λ(ω)2−2ν(ω)[2ν(ω)Mλ(ω)]≤ C∑ω∈G2−ν(ω) ≤ CN,where the second and last steps are consequences, respectively, of Lemma 18.2(i)with $ = ω and β = d− 1 and of Lemma 18.1(i) with α = 1. In both applications,y and $0 have been chosen to be the unit cube, in the root tree and the slope treerespectively.19.2 Proof of Proposition 19.2We are now ready to prove the main Proposition 19.2.Proof. As in the proof of Proposition 19.1, an initial processing of the sum on the lefthand side of (19.2) is needed before embarking on the evaluation of the expectation.208Accordingly, we decompose and simplify the quantity of interest as follows,[∑t1 6=t2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣]2=∑t1 6=t2t′1 6=t′2∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣×∣∣P ∗t′1,σ,R ∩ P∗t′2,σ,R∣∣= S2 + S3 + S4,where for i = 2, 3, 4,Si :=∑Ii∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣×∣∣P ∗t′1,σ,R ∩ P∗t′2,σ,R∣∣, andIi :={I = {(t1, t2); (t′1, t′2)}∣∣∣t1, t2, t′1, t′2 ∈ Q(J), t1 6= t2, t′1 6= t′2,i = number of distinct elements in I}.Without loss of generality, by interchanging the pairs (t1, t2) and (t′1, t′2) if necessary,we may assume that h(D(t1, t2)) ≤ h(D(t′1, t′2)) for all quadruples I ∈ Ii. We willcontinue to make this assumption for the treatment of all three terms Si.The claimed inequality in (19.2) is a consequence of the three main estimatesbelow:EX(S2) ≤ CNM−2R−dJ , (19.7)EX(S3) ≤ CNM−3R−J , and (19.8)EX(S4) ≤ CN2M−4R. (19.9)We will prove (19.9) in full detail, since this clearly makes the primary contributionamong the three terms mentioned above. The other two estimates follow analogousand in fact simpler routes using the machinery developed in Chapters 16 and 17. Weleave their verification to the reader.The configuration type of the quadruple I = {(t1, t2); (t′1, t′2)} of distinct roots, asintroduced in Section 16.6, plays a decisive role in the estimation of (19.9). Recalling209the type definitions from that section, we decompose I4 asI4 =3⊔i=1I4i where I4i :={I ∈ I4∣∣∣I is of type i in thesense of Definition 16.7}.This results in a corresponding decomposition of S4:S4 = S41 + S42 + S43, where S4i =∑I4i∣∣P ∗t1,σ,R ∩ P∗t2,σ,R∣∣×∣∣P ∗t′1,σ,R ∩ P∗t′2,σ,R∣∣.We will prove in Sections 19.2.1–19.2.3 below thatEX[S4i]≤ CN2M−4R for i = 1, 2, 3. (19.10)19.2.1 Expected value of S41We start with S41, simplifying it initially along the same lines as in Proposition 19.1.As before, a summand in S41 is nonzero if and only if the tuple {(t1, t2); (t′1, t′2)} liesin the set{{(t1, t2); (t′1, t′2)} ∈ I41∣∣∣P ∗t1,σ,R ∩ P∗t2,σ,R 6= ∅P ∗t′1,σ,R ∩ P∗t′2,σ,R6= ∅}, (19.11)which in turn is contained in{(t1, t2); (t′1, t′2)} ∈ I41∣∣∣∣∣∃ a unique tuple (v1, v2, v′1, v′2) ∈ Ω4N 3Pt1,v1 ∩ Pt2,v2 ∩ [M−R,M−R+1] × Rd 6= ∅,Pt′1,v′1 ∩ Pt′2,v′2 ∩ [M−R,M−R+1] × Rd 6= ∅,σ(ti) = vi, σ(t′i) = v′i, i = 1, 2. (19.12)210Incorporating this information into the simplification of the sum, we obtainS41 ≤∑1CdM−(d+1)J|σ(t1) − σ(t2)|× CdM−(d+1)J|σ(t′1) − σ(t′2)|︸ ︷︷ ︸Lemma 1.5≤ CM−2(d+1)J∑2T ((t1, v1), (t2, v2))|v1 − v2|× T ((t′1, v′1), (t′2, v′2))|v′1 − v′2|≤ CM−2(d+1)J∑u,u′,zω,ω′,v1δωδω′∑3T ((t1, v1), (t2, v2))T ((t′1, v′1), (t′2, v′2)), (19.13):= S41where the summations∑1 and∑2 range over the root quadruples in (19.11) and(19.12) respectively. The notation T ((t1, v1), (t2, v2)) and δω represent the same quan-tities as they did in Proposition 19.1, with their definitions in (19.6) and (12.8)respectively. Following the same reasoning that led to (19.4), in the last step wehave stratified the sum in terms of the root vertices u = D(t1, t2), u′ = D(t′1, t′2),z = D(u, u′) and the (splitting) slope vertices ω = D(v1, v2), ω′ = D(v′1, v′2),v = D(ω, ω′), so that the summation∑3 takes place over the tube tuples in thecollection E41 = E41[u, u′, z;ω, ω, v; %] defined in (17.11), with % = M−R, C1 = M .We are now in a position to compute the expected value of S41.Lemma 19.4. The estimate in (19.10) holds for i = 1.Proof. Let us refer to the bound S41 on S41 defined by (19.13) that we obtained fromthe preliminary simplification. Assembling the various components of the estimationfrom the previous chapters, the expected value of S41 is estimated as follows,EX(S41)≤ EX(S41)≤ CM−2(d+1)J∑u,u′,zω,ω′,v1ρωρω′︸ ︷︷ ︸Corollary 12.4∑3Pr(σ(ti) = vi, σ(t′i) = v′i, i = 1, 2)211≤ CM−2(d+1)J∑u,u′,zω,ω′,v#(E41)ρωρω′(12)4N−µ(ω,h(u))−µ(ω′,h(u′))−µ(v,h(z))︸ ︷︷ ︸(16.24) from Lemma 16.8≤ CM−2(d+1)J∑u,u′,zω,ω′,v(%2ρωρω′)224N−2(ν(ω)+ν(ω′))M−(d−1)(h(u)+h(u′))+2(d+1)J︸ ︷︷ ︸bound on the size of E41 from Lemma 17.6× 1ρωρω′× 2−4N+µ(ω,h(u))+µ(ω′,h(u′))+µ(v,h(z))≤ CM−4RS∗41, whereS∗41 :=∑ω,ω′,v2−2(ν(ω)+ν(ω′))M−[λ(ω)+λ(ω′)]∑u,u′,z2µ(ω,h(u))+µ(ω′,h(u′))+µ(v,h(z))×M−(d−1)[h(u)+h(u′)].(19.14)It remains to use the appropriate summation results in Chapter 18 to show that S∗41is bounded above by a constant multiple of N2. We start with the inner sum.∑u,u′,zM−(d−1)(h(u)+h(u′))2µ(ω,h(u))+µ(ω′,h(u′))+µ(v,h(z))≤∑z2µ(v,h(z))[∑u⊆zM−(d−1)h(u)2µ(ω,h(u))]︸ ︷︷ ︸apply Lemma 18.2(i), β = d− 1[∑u′⊆zM−(d−1)h(u′)2µ(ω′,h(u′))]︸ ︷︷ ︸apply the same lemma again≤∑z2µ(v,h(z))[M−dh(z)+λ(ω)2ν(ω)][M−dh(z)+λ(ω′)2ν(ω′)]≤ CMλ(ω)+λ(ω′)2ν(ω)+ν(ω′)[∑z2µ(v,h(z))M−2dh(z)]︸ ︷︷ ︸apply Lemma 18.2(iv), h(y)=0≤ CMh(ω)+h(ω′)2ν(ω)+ν(ω′). (19.15)Note that Lemma 18.2(iv) applies with β = 2d since 2Md < M2d for M ≥ 2 andd ≥ 2. Inserting the expression in (19.15) into the inner sum of (19.14), we proceed212to complete the outer sum in S∗41.S∗41 ≤ C∑ω,ω′,v∈GM−λ(ω)−λ(ω′)2−2(ν(ω)+ν(ω′))[Mλ(ω)+λ(ω′)2ν(ω)+ν(ω′)]≤ C∑ω,ω′,v∈G2−ν(ω)−ν(ω′)≤ C∑v∈G[∑ω∈G,ω⊆v2−ν(ω)]︸ ︷︷ ︸apply Lemma 18.1(i), α = 1×[∑ω′∈G,ω′⊆v2−ν(ω′)]︸ ︷︷ ︸same lemma again≤ C∑v∈G[N2−ν(v)]2 ≤ CN2∑v∈G2−2ν(v) ≤ CN2,where at the last step we have again used Lemma 18.1 (i) with α = 2, and ν($0) = 0.This completes the proof of the lemma.19.2.2 Expected value of S42We turn to S42 next. After the usual preliminary simplification similar to that ofS41, we find that S42 is bounded by a sum S42 of the form (19.13), whereS42 := CM−2(d+1)J′∑ 1δωδω′∑3T ((t1, v1), (t2, v2))T ((t′1, v′1), (t′2, v′2)). (19.16)In view of Lemma 16.9 we may assume, after a permutation of (t1, t2) and of (t′1, t′2)if necessary, that the outer sum∑′ in (19.16) is over all vertex tuples (u, u′, t) and(ω, ω′, ϑ) in the root tree and the slope tree respectively, such that u, u′, t lies on asingle ray with u′ ( u, while ω, ω′, ϑ ∈ G(ΩN), ω ∩ ϑ 6= ∅, ω′ ∩ ϑ 6= ∅. The inner sum∑3 in S42 ranges over the collection E42 = E42[u, u′, t;ω, ω′, ϑ; %] defined in (17.12)with the usual % = M−R and C1 = M .Lemma 19.5. The estimate in (19.10) holds for i = 2.Proof. As in Lemma 19.4, the evaluation of the expectation requires a combinationof the appropriate probabilistic estimate from Section 16.2 (specifically Lemma 16.9),213size estimate of E42 from Section 17.3.2 (specifically Lemma 17.8) and the summationresults from Chapter 18. Putting these together, we obtainEX(S42)≤ EX(S42)≤ CM−2(d+1)J′∑ 1ρωρω′∑3Pr(σ(ti) = vi, σ(t′i) = v′i, i = 1, 2)≤ CM−2(d+1)J′∑ #(E42)ρωρω′(12)4N−µ(ω,h(u))−µ(ω′,h(u′))−µ(ϑ,h(t))︸ ︷︷ ︸(16.26) from Lemma 16.9≤ CM−4R[S∗42 + S◦42],where the closed form expressions for S∗42 and S◦42 at the last step are obtained fromthe count on the size of E42 from Lemma 17.8, and reflect the two complementarycases considered therein. To be precise,S∗42 := %−1′∑u′⊆t⊆uρω′ min[%ρω,M−h(t)]M−(d−1)(h(t)+h(u′))(19.17)× 2µ(ω,h(u))+µ(ω′,h(u′))+µ(ϑ,h(t))−m[ω,ω′,ϑ], andS◦42 := %−2′∑t(u′(umin[%ρω,M−h(t)]min[%ρω′ ,M−h(t)]M−2(d−1)h(t) (19.18)× 2µ(ω,h(u))+µ(ω′,h(u′))+µ(ϑ,h(t))−m[ω,ω′,ϑ],where the notation∑′P indicates the subsum of∑′ subject to the additional re-quirement P . These two quantities are estimated via the usual channels. Lemma17.8 places certain restrictions on the spatial location of t, but for a large part of theproof the full strength of these statements will not be needed. For instance, replacingmin(%ρω,M−h(t)) in (19.17) by %ρω, we arrive at the following bound for S∗42:S∗42 ≤′∑u′⊆t⊆uρωρω′M−(d−1)(h(t)+h(u′))2µ(ω,h(u))+µ(ω′,h(u′))+µ(ϑ,h(t))−m[ω,ω′,ϑ]214≤∑ω,ω′,ϑρωρω′2−m[ω,ω′,ϑ]S∗42(inner), (19.19)where the inner expression S∗42(inner) is a sequence of three summations in rootvertices, the computation of each requiring a suitable form of Lemma 18.2. Precisely,S∗42(inner) :=∑u,u′,tu′⊆t⊆uM−(d−1)(h(t)+h(u′))2µ(ω,h(u))+µ(ω′,h(u′))+µ(ϑ,h(t))=∑(t,u)t⊆uM−(d−1)h(t)2µ(ω,h(u))+µ(ϑ,h(t))[∑u′⊆tM−(d−1)h(u′)2µ(ω′,h(u′))]≤ C∑(t,u)t⊆uM−(d−1)h(t)2µ(ω,h(u))+µ(ϑ,h(t))[2ν(ω′)M−dh(t)+λ(ω′)]︸ ︷︷ ︸from Lemma 18.2(i), β = d− 1≤ C2ν(ω′)Mλ(ω′)∑u2µ(ω,h(u))∑t:t⊆uM−(2d−1)h(t)2µ(ϑ,h(t))≤ C2ν(ω′)Mλ(ω′)∑u2µ(ω,h(u))[M−(2d−1)h(u)2ν(ϑ)]︸ ︷︷ ︸from Lemma 18.2(iii), β = 2d− 1≤ C2ν(ω′)+ν(ϑ)Mλ(ω′)∑u2µ(ω,h(u))M−(2d−1)h(u)≤ C2ν(ω′)+ν(ϑ)+ν(ω)Mλ(ω′), (19.20)where the summation in u in the last step also follows from Lemma 18.2(iii), sinceβ = 2d−1 > d. Inserting the estimate (19.20) of S∗42(inner) into (19.19), we proceedto simplify the outer sum. Let us recall from Lemma 17.7 that {ω, ω′, ϑ} can berearranged as {$1, $2, $3} satisfying (17.8), and that m[ω, ω′, ϑ] is defined as in(17.9). Since the definition of m involves two possibilities, we write∑[a] and∑[b]to denote the sum over vertex triples (ω, ω′, ϑ) for which $3 6⊆ $2 and $3 ⊆ $2respectively. This means thatS∗42 ≤ C∑ω,ω′,ϑρωρω′2−m[ω,ω′,ϑ][2ν(ω′)+ν(ϑ)+ν(ω)Mλ(ω′)]215≤ C[[a]∑+[b]∑]ρωρω′2−m[ω,ω′,ϑ][2ν(ω′)+ν(ϑ)+ν(ω)Mλ(ω′)].Using the boundsρω′Mλ(ω′) ≤ C and ρω ≤ CM−λ(ω) ≤ CM−λ($1),the estimation is completed as follows,[a]∑ρωρω′2−m[ω,ω′,ϑ][2ν(ω′)+ν(ϑ)+ν(ω)Mλ(ω′)]≤ C∑$1M−λ($1)2ν($1)[∑$2$2⊆$12−ν($2)]×[∑$3$3⊆$12−ν($3)]≤ C∑$1M−λ($1)2ν($1)(N2−ν($1))2︸ ︷︷ ︸Lemma 18.1 (i) twice≤ CN2∑$1M−λ($1)2−ν($1)︸ ︷︷ ︸apply Lemma 18.1(ii)≤ CN2.The same bound holds for∑[b], and is proved along similar lines:[b]∑ρωρω′2−m[ω,ω′,ϑ][2ν(ω′)+ν(ϑ)+ν(ω)Mλ(ω′)]≤ C∑$1,$2$2⊆$1M−λ($1)∑$3$3⊆$22−ν($3) ≤ C∑$1,$2$2⊆$1M−λ($1)[N2−ν($2)]︸ ︷︷ ︸Lemma 18.1 (i)≤ CN∑$1M−λ($1)∑$2⊆$12−ν($2) ≤ CN∑$1M−λ($1)[N2−ν($1)]︸ ︷︷ ︸Lemma 18.1 (i)≤ CN2∑$1M−λ($1)2−ν($1)︸ ︷︷ ︸apply Lemma 18.1(ii)≤ CN2.This completes the estimation of S∗42.We briefly remark on the analysis of S◦42. For d ≥ 3, replacing the minima in216(19.18) by the trivial bounds %ρω and %ρω′ results in an expression analogous to thatof S∗42:S◦42 ≤′∑t(u′(uρωρω′M−2(d−1)h(t)2µ(ω,h(u))+µ(ω′,h(u′))+µ(ϑ,h(t))−m[ω,ω′,ϑ].This term is estimated exactly the same way as S∗42, since Lemma 18.2(iii) appliesas before with β = 2(d − 1) > d per our choice of d. The bound obtained is aconstant multiple of N . These details are omitted to avoid repetition. We onlypresent the case d = 2, where Lemma 18.2 does not give the desired consequence,and the treatment of which exhibits a slight departure from the norm so far. Ford = 2, inserting the bound min(%ρω,M−h(t)) ≤ %ρω into (19.18) yieldsS◦42 ≤′∑t(u′(u2µ(ω,h(u))+µ(ω′,h(u′))+µ(ϑ,h(t))−m[ω,ω′,ϑ]×ρωρω′M−2h(t) if M−h(t) ≥ %ρω′ ,%−1ρωM−3h(t) if M−h(t) < %ρω′ .(19.21)Further, Lemma 17.8(ii) prescribes that t cannot be arbitrarily placed inside u′, butmust lie within the union of at most 2M thin rectangles of dimension %ρω′ ×M−h(u′)each. Using this information, we sum the expression (19.21) in t as follows: if∑1and∑2 denote the summations in t with t ⊆ u′ and E42 6= ∅ subject to the conditionsM−h(t) ≥ %ρω′ and M−h(t) < %ρω′ respectively, thenρωρω′∑1M−2h(t)2µ(ϑ,h(t)) + %−1ρω∑2M−3h(t)2µ(ϑ,h(t))≤ Cρωρω′[2ν(ϑ)M−2h(u′)]+ C%−1ρω[2ν(ϑ)M−h(u′)(%ρω′)min[%ρω′ ,M−h(u′)]]≤ Cρωρω′2ν(ϑ)M−2h(u′),where both sums have been evaluated using Lemma 18.3 with d = 2, r = 1, $ = ϑ,β =  = M−h(u′) and γ = %ρω′ . In particular,∑1 appeals to part (i) of this lemma217with α = 2 while∑2 uses part (ii) with α = 3. Incorporating this into (19.21), wefind thatS◦42 ≤∑ω,ω′,ϑρωρω′2ν(ϑ)−m[ω,ω′,ϑ]S◦42(inner), where (19.22)S◦42(inner) :=∑u2µ(ω,h(u))∑u′⊆uM−2h(u′)2µ(ω′,h(u′))︸ ︷︷ ︸apply Lemma 18.2(ii)(19.23)≤ C2ν($2)λ($2)∑uM−2h(u)2µ(ω,h(u))︸ ︷︷ ︸apply the same lemma again(19.24)≤ C2ν($2)+ν($1)λ($2)λ($1), (19.25)We pause for a moment to explain these steps. In the first application of Lemma18.2(ii) in (19.23) above we have used, in addition to h(u′) ≤ λ(ω′), the fact thath(u′) = h(D(t′1, t′2)) ≤ h(t) = h(D(t2, t′2)) ≤ λ(D(v2, v′2)) = λ(ϑ),which is a consequence of weak stickiness. Since one of ω′ and ϑ is contained in theother, this implies that µ(ω′, h(u′)) = µ(ϑ, h(u′)). Hence Lemma 18.2(ii), appliedonce with $ = ω′ and again with $ = ϑ, yields∑u′⊆uM−2h(u′)2µ(ω′,h(u′)) ≤ CM−2h(u) min[2ν(ϑ)λ(ϑ), 2ν(ω′)λ(ω′)]≤ Cλ($2)2ν($2)M−2h(u).The second application of Lemma 18.2(ii) in (19.24) uses a similar argument relyingon the fact that h(u) ≤ λ($1). Inserting (19.25) into (19.22), the estimation of S◦42can now be completed in the same way as for S∗42:S◦42 ≤ C∑ω,ω′,ϑρωρω′λ($2)λ($1)2ν(ϑ)+ν($1)+ν($2)−m[ω,ω′,ϑ]218≤ C∑ω,ω′,ϑM−λ($1)−λ($2)λ($1)λ($2)2ν(ϑ)+ν($1)+ν($2)−m[ω,ω′,ϑ]≤ C[a]∑M−12λ($1)−12λ($2)2−ν($3)−ν($2)+ν($1) +[b]∑M−12λ($1)−12λ($2)2−ν($3)≤ CN,where the symbols∑[a] and∑[b] carry the same meaning as they did in the esti-mation of S∗42 and the last step involves several summations all of which have usedappropriate parts of Lemma 18.1. The estimation of S42 is complete.19.2.3 Expected value of S43Lemma 19.6. The estimate in (19.10) holds for i = 3.Proof. After the usual initial processing of S43 which we omit, we reduce to thefollowing estimate:EX(S43)≤ CM−2(d+1)J′∑ #(E43)ρωρω′(12)4N−µ(ω,h(u))−µ(ϑ,h(s1))−µ(ϑ2,h(s2))≤ C′∑(ρωρω′)−1(%ρω′)2M−2(d−1)h(s2)2∏i=1[min[%ρω,M−h(si)]]× 2−m[ω,ω′,ϑ1,ϑ2]+µ(ω,h(u))+µ(ϑ1,h(s1))+µ(ϑ2,h(s2))≤ CM−4R[S∗43 + S◦43],where∑′ denotes the sum over all tuples (u, s1, s2) in the root tree and (ω, ω′, ϑ1, ϑ2)in the slope tree such that s1, s2 ⊆ u, h(u) ≤ h(s1) ≤ h(s2), ρω ≤ ρω′ and forwhich E43 is nonempty. The second inequality displayed above uses the estimateon #(E43) obtained in Lemma 17.9, with an additional simplification resulting frommin(%ρω′ ,M−h(si)) ≤ %ρω′ . The quantities S∗43 and S◦43 refer to the subsum of∑′under the additional constraints of M−h(s1) ≥ %ρω and M−h(s1) < %ρω respectively.219ThusS∗43 = %−1′∑M−h(s1)≥%ρωρω′ min[%ρω,M−h(s2)]M−2(d−1)h(s2)× 2−m[ω,ω′,ϑ1,ϑ2]+µ(ω,h(u))+µ(ϑ1,h(s1))+µ(ϑ2,h(s2))=: %−1∑ω,ω′,ϑ1,ϑ2ρω′2−m[ω,ω′,ϑ1,ϑ2]S∗43(inner), and (19.26)S◦43 = %−2′∑M−h(s1)<%ρωρω′ρωmin[%ρω,M−h(s2)]M−2(d−1)h(s2)−h(s1)× 2−m[ω,ω′,ϑ1,ϑ2]+µ(ω,h(u))+µ(ϑ1,h(s1))+µ(ϑ2,h(s2))=: %−2∑ω,ω′,ϑ1,ϑ2ρω′ρω2−m[ω,ω′,ϑ1,ϑ2]S◦43(inner). (19.27)For the purpose of simplifying S∗43(inner), we recall from Lemma 17.9(ii) that s2 ( uhas sidelength no more than M−h(s1), and moreover, is constrained to lie in the unionof at most 2dM parallelepipeds with (d− 1) long directions and one short direction,of dimensions M−h(s1) and %ρω respectively. Denoting by∑∗s2 the summation overall such cubes s2, we find that∗∑s22µ(ϑ2,h(s2))M−2(d−1)h(s2) min[%ρω,M−h(s2)]≤ %ρω∗∑M−h(s2)≥%ρωM−2(d−1)h(s2)2µ(ϑ2,h(s2)) +∗∑M−h(s2)<%ρωM−(2d−1)h(s2)2µ(ϑ2,h(s2))≤ %ρωs+ + s−≤ C[%ρω2ν(ϑ2)M−2(d−1)h(s1) + 2ν(ϑ2)(%ρω)dM−(d−1)h(s1)]≤ C%ρω2ν(ϑ2)M−2(d−1)h(s1), (19.28)where s± are defined as in (18.1), and estimated according to Lemma 18.3, with theparameters being set at  = β = M−h(s1), γ = %ρω, $ = ϑ2 for both. The value of αis 2(d− 1) for s+ and (2d− 1) for s−. A similar argument applies for the summation220in s1 with M−h(s1) ≥ %ρω. According to Lemma 17.9(i), s1 has to lie in u and withina distance at most C∆ from the boundary of some child of u. Hence the range ofs1 lies within the union of at most dM parallelepipeds, each of dimension M−h(u) in(d− 1) directions and C∆ in the remaining one. Denoting by∑∗s1 the relevant sum,and applying Lemma 18.3 again with α = 2(d− 1), r = 1,  = β = M−h(u), γ = %ρω,$ = ϑ1,∗∑s1M−2(d−1)h(s1)2µ(ϑ1,h(s1)) ≤ s+ ≤ 2ν(ϑ1)M−2(d−1)h(u). (19.29)Inserting the estimates (19.28) and (19.29), we arrive at the following bound onS∗43(inner):S∗43(inner) =∑u∗∑s12µ(ω,h(u))+µ(ϑ1,h(s1))×[∗∑s22µ(ϑ2,h(s2))M−2(d−1)h(s2) min[%ρω,M−h(s2)]]≤ C∑u,s12µ(ω,h(u))+µ(ϑ1,h(s1))[2ν(ϑ2)%ρωM−2(d−1)h(s1)]≤ %ρω2ν(ϑ2)∑u2µ(ω,h(u))∗∑s1M−2(d−1)h(s1)2µ(ϑ1,h(s1))≤ %ρω2ν(ϑ2)∑u2µ(ω,h(u))[M−2(d−1)h(u)2ν(ϑ1)]≤ %ρω2ν(ϑ2)+ν(ϑ1)∑u2µ(ω,h(u))M−2(d−1)h(u)≤ %ρω2ν(ϑ2)+ν(ϑ1)+ν($1)λ($1), (19.30)where $1 is the youngest common ancestor of ω, ω′, ϑ1, ϑ2, and hence λ($1) ≥ h(u).The last estimate follows from Lemma 18.2, invoking part (iii) if d ≥ 3 and part(i)if d = 2. An analogous sequence of steps, the details of which are left to the reader,can be executed to estimate S◦43(inner), the only distinction being that the spacerestrictions are now dictated by Lemma 17.9(iii), so that the summation in s2 invokesLemma 18.3 with r = d−1, β = %min(M−λ(ω),M−λ(ω′)), γ = M−h(s1). The outcome221of this is thatS◦43(inner) ≤ %2ρω min(M−λ(ω),M−λ(ω′))2ν(ϑ2)+ν(ϑ1)+ν($1)λ($1). (19.31)Substituting (19.30) into (19.26) and (19.31) into (19.27) leads to the following sim-pler sum over slope vertices:S∗43 + S◦43 ≤ C∑ω,ω′,ϑ1,ϑ2M−λ(ω)−λ(ω′)2−m[ω,ω′,ϑ1,ϑ2]+ν($1)+ν(ϑ1)+ν(ϑ2).In order to complete the summation, let us recall that the sum, ostensibly overfour parameters, in fact ranges over at most three vertices {$1, $2, $3}, which is arearrangement of the quadruple {ω, ω′, ϑ1, ϑ2} satisfying (17.8). However, it is notapriori possible to assign a unique correspondence between these two sets of vertices.Indeed, as already indicated in the last paragraph of Chapter 16, the configurationtype of the slopes (which does not in general mimic the configuration type of theroots) dictates which vertex or vertices of the quadruple {ω, ω′, ϑ1, ϑ2} represents $iafter the rearrangement. A careful analysis of the possible structures of ω, ω′, ϑ1, ϑ2,as depicted in Figure 19.1, shows thatM−λ(ω)−λ(ω′)λ($1)2−m[ω,ω′,ϑ1,ϑ2]+ν($1)+ν(ϑ1)+ν(ϑ2)≤M−2λ($1)λ($1) ×2−ν($3)−ν($2)+ν($1) if $3 6⊆ $22−ν($3) if $3 ⊆ $2.The expression on the right hand side is of the type already considered in the esti-mation of S∗42 and S◦42. In particular, it is summable in $1, $2, $3 using repeatedapplications of Lemma 18.1 and yields the desired bound of CN2.222ϑ1 = ϑ2 = $1ω = $2ω′ = $3v1 v2 v′1 v′2(1)ω = ϑ1 = $1ω′ = $2ϑ2 = $3v1 v′1 v′2 v2(3)ω = ϑ1 = $1ϑ2 = $2ω′ = $3v1 v2 v′1 v′2(2)Figure 19.1: A partial list of 4-slope configurations for 4 roots of type 3, withdistinct {$1, $2, $3}. Other configurations (where partial coincidences may arise)are possible after permutation of {v1, v′1, v2, v′2} in these diagrams.223Chapter 20Future workIn this final chapter, we discuss some potential ideas for future work to come more orless directly out of the work presented in the current document. Some of these ideashave yet to be seriously considered, while others, specifically those in Section 20.2are presently being developed in collaboration with Dimitrios Karslidis and MalabikaPramanik. All material presented in this chapter should be considered work inprogress and not established fact.20.1 Maximal functions over other collections ofsticky objectsWe briefly mentioned at the end of Section 1.5 that the notion of stickiness is notunique to collections of tubes. Indeed, curves and spheres are among two of themost well studied objects that can also be grouped into sticky collections; see [8],[29], [40], [30] for example. Besicovitch and Rado [8] proved that there exist measurezero sets in Rd that contain a sphere of every radius (Kinney [29] proved the sameconcurrently in two dimensions). Following the same argument used to deduce theunboundedness of the Kakeya-Nikodym maximal operator, presented in Section 1.3,224the existence of such sets means that the circular maximal function defined asM circf(x) := supr∈R+1|C(x, r)|∫C(x,r)|f(y)|dy, (20.1)must be unbounded as an operator on Lp(Rd) for p ∈ [1,∞). Here, C(x, r) denotesthe sphere centred at x with radius r. Naturally, we then consider the restrictedcircular maximal function defined asM circδ f(x) := supr∈R+1|Cδ(x, r)|∫Cδ(x,r)|f(y)|dy, (20.2)where Cδ(x, r) denotes the δ-thickening of the circle C(x, r). Kolasa and Wolff derivedδ-dependent bounds on the size of the norm of an equivalent operator. Interestingly,their bounds turn out to be optimal in dimensions d ≥ 3 [30].Like their classical counterparts, zero measure circular Kakeya sets seem to ex-hibit a stickiness quality in that many spheres positioned close to each other in spacemust have comparable radii. This motivates the notion of circular Kakeya-type sets.Definition 20.1. Fix a set of radii Φ ⊆ R+. If for some fixed constant A0 ≥ 1 andany choice of integer N ≥ 1, there exist- a number 0 < δN  1, δN ↘ 0 as N ↗∞, and- a collection of spheres {S(N)t } with radii in Φ and thickness at most δNobeyinglimN→∞|E∗N(A0)||EN |= ∞, with EN :=⋃tS(N)t , E∗N(A0) :=⋃tA0S(N)t , (20.3)then we say that Φ admits circular Kakeya-type sets. Here, A0S(N)t denotes thesphere with the same centre and thickness as S(N)t but A0 times the radius.The existence of such sets for a given Φ ⊆ R+ should imply the Lp-unboundednessof a corresponding maximal operator analogous to (20.1), where the supremum is225now taken over all radii r ∈ Φ. While the geometry is considerably different in thiscontext, it is perhaps reasonable to hope that the construction of circular Kakeya-type sets could be approached using some of the ideas developed in the work ofBateman and Katz, as well as in this document.Beside for spheres, it may also be worthwhile to consider if the techniques de-veloped in this document can be applied to the construction of Kakeya-type setsover flat slabs. Flat slabs are essentially thickened circles and can be defined as theintersection of a hyperplane with a circular cylinder whose principal axis sits normalto the hyperplane. We say that such a slab is oriented in the direction given by thisnormal vector.Definition 20.2. Fix a set of directions Ω ⊆ Sd. If for some fixed constant A0 ≥ 1and any choice of integer N ≥ 1, there exist- a number 0 < δN  1, δN ↘ 0 as N ↗∞, and- a collection of δN -thickened slabs {R(N)t } with orientations lying inside Ω and di-ameter at least 1obeyinglimN→∞|E∗N(A0)||EN |= ∞, with EN :=⋃tR(N)t , E∗N(A0) :=⋃tA0R(N)t , (20.4)then we say that Ω admits Kakeya-type sets over slabs. Here, A0R(N)t denotes the slabwith the same centre, orientation and thickness as R(N)t but A0 times its diameter.Again, the existence of such sets would have implications for the Lp behaviourof a corresponding maximal operator. How much the techniques discussed in thisdocument can be used to study these and other like sets is not currently known, butthe potential for further applications appears viable.22620.2 A characterization of the Lp(Rd+1)-boundednessof directional maximal operators over an ar-bitrary set of directionsUltimately, one would like to establish a necessary and sufficient condition for theboundedness of directional maximal operators in general Euclidean space. One wouldhope that this goal could be recast to completely characterize those direction setsΩ that admit Kakeya-type sets according to Definition 1.1. We aim to combine theresult of Theorem 1.3 with others from the literature, most notably [3], [1], [39], toobtain the following necessary and sufficient conditions for Kakeya-type sets to exist.Conjecture 20.3. (Kroc, Pramanik) For any d ≥ 1, the following are equivalent:(1) The direction set Ω ⊆ Sd is sublacunary in the sense of Definition 2.7.(2) The set of directions Ω admits Kakeya-type sets in the sense of Definition 1.1.(3) The maximal operators DΩ and MΩ defined in (1.2) and (1.3) respectively areunbounded on Lp(Rd+1) for every p ∈ (1,∞).To clarify, the implication (3) =⇒ (1) for d = 1 is in [1], expanding on thework started in [37], [41], [10], [2]. For d ≥ 2, our notion of lacunarity paired withthe result of Parcet and Rogers [39] suggests a possible bridge, although substantialdetails remain to be resolved to fully connect their work with ours; see below. Theproof of (1) =⇒ (2) is sketched in [3] for d = 1. Theorem 1.3 is exactly theimplication (1) =⇒ (2) for all d ≥ 1. The implication (2) =⇒ (3) is easy and isestablished in the argument presented in the paragraph of (1.9) in all dimensions.Some of the implications above are known to admit stronger variants. For in-stance, (2) implies (3) even when p = 1, as the argument leading to (1.9) shows.Further, it does not appear to be necessary to know that the operator DΩ is un-bounded on all Lp(Rd+1), p ∈ (1,∞), in order to conclude that Ω is sublacunary. Wewill sketch a possible path next in Section 20.2.1 toward the weaker requirement(3’) The maximal operator DΩ is unbounded on Lp(Rd+1) for some p ∈ (1,∞),227which suffices to establish (1). Thus DΩ enjoys an interesting dichotomy in that itis either bounded on all or none of the Lebesgue spaces Lp with p ∈ (1,∞).20.2.1 Boundedness of directional maximal operators, sketchIn this subsection, we provide a sketch of the implication (3) =⇒ (1) of Theo-rem 20.3, relying heavily on the result of Parcet and Rogers [39]. Let us recall from(1.2) and (1.3) the relevant definitions.Conjecture 20.4. Given positive integers N,R, a positive constant λ < 1 and anyexponent p ∈ (1,∞], there exists a positive finite constant Cp = Cp(N, λ,R) with thefollowing property. Any admissible lacunary direction set Ω ⊆ Rd+1 of finite orderthat obeys Definition 2.7(i) with the prescribed values of N , λ and R also satisfies||MΩ||p→p ≤ Cp and ||DΩ||p→p ≤ Cp. (20.5)Proof. (Sketch) We first argue that the boundedness of DΩ on any Lp(Rd+1) impliesthe same for MΩ. Without loss of generality, we may assume that Ω ⊆ (−, )d×{1}for some small constant  > 0. Let us define for any x ∈ Rd+1 the vectorsvj(x) = xd+1ej − xjed+1, 1 ≤ j ≤ d,where {e1, · · · ed+1} denotes the canonical orthonormal basis in Rd+1. For ω =(ω1, · · · , ωd, 1) ∈ Ω, the collection {v1(ω), · · · , vd(ω)} spans ω⊥. ThenMΩf(x) ≤ Cd supω∈Ωsup0<r≤h1rdh∫|t|≤h|s|≤r∣∣f(x− tω −d∑j=1sjvj(ω))∣∣ dt ds≤ Cd supω∈Ωsupr>01rd∫DΩf(x−d∑j=1sjvj(ω))ds≤ Cd supω∈Ωsupr>01rd−1∫DΩ1 ◦DΩf(x−d∑j=2sjvj(ω))ds2 · · · , dsd≤ · · · ≤ CdDΩd ◦DΩd−1 ◦ · · · ◦DΩ1 ◦DΩf(x),228where Ωj = {vj(ω) : ω ∈ Ω}, 1 ≤ j ≤ d. The relationvj(ω) · ξvj(ω) · η= ω · vj(ξ)ω · vj(η)for all ξ, η ∈ Rimplies that if Ω is admissible lacunary of order at most N as a direction set, thenso is Ωj for every j. Thus a bound on the Lp operator norm of MΩ would follow ifthe second conclusion (for directional maximal operators) in (20.5) is known to holdfor all such direction sets. We will henceforth concentrate only on DΩ, with Ω beingadmissible lacunary of finite order N .As mentioned before, the Lp-boundedness of DΩ can be seen to follow from themain result in [39], modulo considerable connecting detail. We sketch these detailsnow and then indicate where more rigour is required.The proof is by induction on the order of lacunarity N . The initializing stepN = 0 is a consequence of the one-dimensional Hardy-Littlewood maximal theorem.To set up the induction step, we observe that||DΩ||p→p = ||DT (Ω)||p→p for all p ∈ [1,∞]if T (Ω) is one of the following two types:- T (Ω) = {Aω : ω ∈ Ω} for any nonsingular linear transformation A, or- T (Ω) = {cωω : ω ∈ Ω} for any collection on nonzero scalars {cω}.Clearly, Ω and T (Ω) also have the same order of lacunarity as direction sets. We saythat Ω′ is a representative of Ω if there is a finite sequence of transformations T ofthe types described above that maps Ω to Ω′.Set Σ := {(j, k) : 1 ≤ j < k ≤ d + 1}. We will prove in Lemma 20.5 below thatafter a decomposition into at most C(d,R) pieces, it is possible to find a represen-tative Ω′ of the direction set Ω obeying certain desirable properties. Specifically, forany σ = (j, k) ∈ Σ, definepiσ(Ω′) :={ωkωj: (ω1, · · · , ωd+1) ∈ Ω′},229which is the projection of CΩ′ ∩ {ωj = 1} onto the kth coordinate axis. Then thereexists an integer 0 ≤ Nσ ≤ N such that piσ(Ω′) ∈ Λ(Nσ;λ) \ Λ(Nσ − 1;λ). SetΣ∗ = Σ∗(Ω′) := {σ ∈ Σ : Nσ ≥ 1}. In order to apply the main result of [39], werequire that for any σ ∈ Σ∗,the special point of piσ(Ω′), as defined in Definition 2.1, is 0. (20.6)For σ ∈ Σ∗, let {θσ,i : i ∈ Z} denote a monotone decreasing non-negative lacunarysequence with lacunary constant ≤ λ that converges to 0 and ∞ as i → ∞ andi→ −∞ respectively, and which serves as a special sequence for piσ(Ω′). DefineΩ′σ,i :={ω = (ω1, · · · , ωd+1) ∈ Ω′ : θσ,i+1 <∣∣∣∣ωkωj∣∣∣∣≤ θσ,i}. (20.7)This puts us in the framework of [39], where the authors prove that||DΩ||p→p = ||DΩ′||p→p ≤ C supσ∈Σ∗supi≥1||DΩ′σ,i||p→p.(In fact, [39] addresses the generic and nontrivial case of Σ∗ = Σ, but the proof goesthrough with trivial modifications after a reduction to lower dimensions even whenΣ∗ ( Σ). Lemma 20.5 ensures that each set Ω′σ,i is lacunary of order N − 1 as adirection set, allowing us to carry the induction forward.Lemma 20.5. Given integers N,R ≥ 1, and a constant 0 < λ < 1, let Ω0 bea direction set that is admissible lacunary of finite order, obeying Definition 2.7(i)with these values of N,R, λ. Then there exists a decomposition of Ω0 into at mostC(d,R) subsets of directions {Ω}, each of which has the following property: thereexists a representative Ω′ of Ω such that(i) Ω′ is contained in a hyperplane at unit distance from the origin,(ii) the condition (20.6) holds,(iii) for any choice of σ ∈ Σ∗ and i ∈ Z, the direction set Ω′σ,i given by (20.7) islacunary of order at most N − 1.230Proof. (Sketch) We use a generic linear transformation A0 of Rd+1 to fix an ambientcoordinate system, in which we represent Ω0. By decomposing Ω0 into at mostRd(d−1)/2pieces {Ω} if necessary, we may assume that piσ(Ω) ∈ Λ(N ;λ) for everyσ ∈ Σ. We now fix such a direction set Ω, and aim to apply on it a sequence oftransformations of the types mentioned in the proof of Theorem 20.2.1, eventuallyreaching a representative that meets the specified criteria.After a splitting into d! subsets and permuting coordinates so that Ω ⊆ {|x1| ≥|x2| ≥ · · · ≥ |xd+1|}, we start with the representative Ω[0] = CΩ∩{x1 = 1} of Ω. If akdenotes the special point of pi(1,k)(Ω[0]), we set Ω[1] = T (Ω[0]), where T is the lineartransformationT (x1, · · · , xd+1) = (x1, x2 − a2x1, · · · , xd+1 − ad+1x1).Then piσ(Ω[1]) has 0 as its special point for all σ of the form σ = (1, k). We setΓ[1]g = {1} and Γ[1]b = {2, 3, · · · , (d + 1)}. These collections represent the subsets of“good” and “bad” coordinates respectively at the end of the first step.Inductively, we define representatives Ω[m] of Ω such that for each 1 ≤ m ≤ d,the disjoint collections of good and bad indicesΓ[m]g = {1, · · · ,m} and Γ[m]b = {m+ 1, · · · , d+ 1}have the following significance:- Ω[m] ⊆ {ω ∈ Rd+1 : ωm = 1}.- For any σ = (j, k) ∈ Σ with j, k ∈ Γ[m]g , the special point of the set piσ(Ω[m]) is 0.- For σ = (j, k) ∈ Σ with j ∈ Γ[m]g , k ∈ Γ[m]b , the set piσ(Ω[m]) has 0 as its specialpoint as well.Given a representative Ω[m] with these properties, we permute coordinates so thatafter a finite decomposition Ω[m] ⊆ {|ωm+1| ≥ · · · ≥ |ωd+1|}. The scaling transfor-231mation(ω1, · · · , ωm = 1, · · · , ωd+1) ∈ Ω[m] 7→ (ζ1, · · · , ζd+1) ∈ Θ, where ζj =ωjωm+1,leads to the inclusion Θ ⊆ {ζm+1 = 1}. On the other hand, by the inductionhypothesis piσ(Θ) = piσ(Ω[m]) has 0 as its special point for every σ = (j, k) ∈ Σ,1 ≤ j, k ≤ m + 1. We next apply the linear transformation Ω[m+1] = T (Θ) whereη = T (ζ) is given byηk =ζk if k ≤ m+ 1,ζk − bkζm+1 if k > m+ 1,and bk denotes the special point of piσ(Θ) for σ = (m + 1, k) ∈ Σ. For 1 ≤ j ≤m + 1 < k ≤ d + 1, the induction hypothesis again yields that the sets {η−1j :(η1, · · · , ηd+1) ∈ Ω[m+1]} and {ηk : (η1, · · · , ηd+1) ∈ Ω[m+1]} both have their specialpoint at zero. The definition of finite order lacunarity of Ω then implies that thesame conclusion continues to hold true for piσ(Ω[m+1]), with σ = (j, k). Thus Ω[m+1]satisfies the requirements listed above with m replaced by (m+ 1).Proceeding in this manner for d steps, we arrive at a representation Ω′ = Ω[d] ofΩ, for which Γ[d]g = {1, · · · , d}, and which therefore verifies the requirements (i) and(ii) of the lemma. For Ω′σ,i defined as in (20.7), we claim that piσ(Ω′σ,i) ∈ Λ(N −1;λ).Let us recall that Ω′σ,i is lacunary of finite order as a direction set (being the subsetof a representative of Ω), and that piσ(Ω′σ,i) is the projection of CΩ′σ,i ∩ {ζj = 1} ontothe ζk-axis. Hence for a generic choice of coordinates based on the selection of A0 atthe beginning of this proof (see also the last remark on page 31), Ω′σ,i is lacunary asa direction set of order at most N − 1.23220.2.2 Boundedness of directional maximal operators, a de-tailed exampleThe main gap that remains to be filled in the previous subsection’s sketch is theclaim that the projections onto the coordinate axes of any hyperplane {xj = 1}of the sets Ω′σ,i are lacunary of order strictly less than N . Such a statement isabsolutely necessary if we hope to verify boundedness via an inductive argument.In this subsection, we will indicate how to justify such a claim in three dimensionswhen N = 2. This will already indicate where much of the additional work lies.Suppose Ω is admissible lacunary of order 2 as a direction set in R3; denote thisby Ω ∈ ∆(2, λ;R) so that Ω is coverable by R lacunary direction sets of order 2.Furthermore, suppose that CΩ∩{x1 = 1} has the property that ∃ a ∈ {x1 = 1} suchthat if L is a line in {x1 = 1} passing through a, then piL(CΩ ∩ {x1 = 1}) =⋃Ri=1 Ui,with Ui ∈ Λ(2, λ) and special point a for all i. Without loss of generality, we mayassume a = (1, 0, 0). We will say that such a projection obeying the above propertyis a member of Λ(2, λ;R) relative to the origin.Write CΩ ∩ {x1 = 1} = {(1, ω2, ω3) ∈ Ω}. Then using the projection notationof the previous subsection, we see that piσ(CΩ ∩ {x1 = 1}) ∈ Λ(2, λ;R) relative tothe origin for σ = (1, 2) and σ = (1, 3). In order to apply the result of Parcet andRogers, we must also be able to conclude the same when σ = (3, 2). It is true thatpi(3,2)(CΩ ∩ {x1 = 1}) ={ω2ω3: (1, ω2, ω3) ∈ Ω}∈ Λ(2, λ;R)since Ω ∈ ∆(2, λ;R) by hypothesis, but the lacunarity of this projection is notnecessarily relative to a single point. That is, a priori, there is nothing to rule outthe possibility that the projection is only coverable by R different Euclidean sets,lacunary of order 2, all with potentially different special points.This requires a decomposition of the original direction set Ω. We have that{ω2ω3: (1, ω2, ω3) ∈ Ω}=R⋃i=1Vi,233with Vi ∈ Λ(2, λ) and ci the special point of Vi. Split Ω into R pieces Ωi such thatCΩi ∩ {x1 = 1} ={(1, ω2, ω3) ∈ Ωi :ω2ω3∈ Vi}.Henceforth, we fix i (which fixes Vi and ci) and rename Ωi as Ω.Now we apply a linear transformation T to Ω, T (Ω) := {(1, ω2 − cω3, ω3) :(1, ω2, ω3) ∈ Ω}. By hypothesis, we have {ω2 − cω3} ∈ Λ(2, λ;R). Consequently,piσ(T (Ω)) ∈ Λ(2, λ;R) relative to the origin for all σ ∈ {(1, 2), (1, 3), (3, 2)}.We are now in a position to apply the result of [39]:||MΩ||p→p = ||MT (Ω)||p→p . supσsupi≥1||MΩ′σ,i||p→p, (20.8)where Ω′σ,i is as defined in (20.7). The symmetry of the following argument will allowus to fix σ = (1, 2) and concentrate on the quantity supi≥1 ||MΩ′σ,i ||p→p. We claimthe following:∃ C = C(R) <∞ such that Ω′(1,2),i ∈ Λ(1, λ;C) ∀ i as a Euclidean set. (20.9)What we want of course is for the angular sectors Ω′(1,2),i to actually be admissiblelacunary of order 1 as direction sets, but it seems that this is too much to hope forafter only a single application of the inequality in [39]. However, there is nothingto prevent us from decomposing these sectors further and applying inequality (20.8)again.Fix i ≥ 1. Assuming (20.9), we note that both {ω′2} ∈ Λ(1, λ;C) and {ω′3} ∈Λ(1, λ;C) where Ω′(1,2),i = {(1, ω′2, ω′3)}. By decomposing Ω′(1,2),i into at most C2many pieces {Ω′′(1,2),i}, we can ensure that {ω′2} ∈ Λ(1, λ) with special point αi, and{ω′3} ∈ Λ(1, λ) with special point βi. Applying another linear transformation, wehave the setΩ′′′(1,2),i = {(1, ω′2 − αi, ω′3 − βi) : (1, ω′2, ω′3) ∈ Ω′′(1,2),i}.This allows us to conclude that the projections have lower order lacunarity with234special points at the origin for σ = (1, 2) and σ = (1, 3), but when σ = (3, 2) weknow only thatω′2 − αiω′3 − βi∈ Λ(2, λ;R).Decompose Ω′′′(1,2),i into at most R pieces to ensure thatω′2 − αiω′3 − βihas a unique specialpoint γi. After another linear transformation, we define the following set:˜Ω(1,2),i := {(1, (ω′2 − αi) − γi(ω′3 − βi), ω′3 − βi) : (1, ω′2, ω′3) ∈ Ω′′(1,2),i}. (20.10)Now by construction, we see that piσ(˜Ω(1,2),i) ∈ Λ(2, λ;R) relative to the origin forall σ ∈ {(1, 2), (1, 3), (3, 2)}. Thus, the result of Parcet and Rogers applies and weonce again apply inequality (20.8):||M˜Ω(1,2),i ||p→p . supσsupj≥1||M˜Ωσ,j ||p→p,where we have suppressed the fixed subscripts (1, 2) and i on the right hand side andonce again ˜Ωσ,j is defined as in (20.7).Now for σ = (1, 2), the dimension of the direction set ˜Ωσ,j drops, allowing us tothen induct on the dimension. Indeed, writing ˜Ω(1,2),j = {(1, x2, x3)}, we see thatthere can exist no more than R many distinct values of x2. Of course, there mayexist many values of x3 so that (1, x2, x3) ∈ ˜Ω(1,2),j for one particular x2, but thenthe dimension of the overall set drops. The same holds if σ = (1, 3).For σ = (3, 2), we claim that since {x2}, {x3} ∈ Λ(1, λ) with special point theorigin, and{x2x3}∈ Λ(1, λ), possibly after a finite decomposition, then ˜Ω(3,2),j ∈∆(1, λ;R). 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