Restriction Theorems and Salem SetsbyKyle David HambrookA thesis submitted in partial fulfillment of the requirements forthe degree ofDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)The University of British Columbia(Vancouver)July 2015c© Kyle David Hambrook, 2015AbstractIn the first part of this thesis, I prove the sharpness of the exponent range in theL2 Fourier restriction theorem due to Mockenhaupt [37] and Mitsis [35] (with theendpoint due to Bak and Seeger [1]) for measures on R. The proof is based on arandom Cantor-type construction of Salem sets due to Laba and Pramanik [31]. Thekey new idea is to embed in the Salem set a small deterministic Cantor set thatdisrupts the restriction estimate for the natural measure on the Salem set but doesnot disrupt the measure’s Fourier decay.In the second part of this thesis, I prove a lower bound on the Fourier dimensionofE(Q,Ψ, θ) = {x ∈ R : ‖qx− θ‖ ≤ Ψ(q) for infinitely many q ∈ Q},where Q is an infinite subset of Z, Ψ : Z → [0,∞), and θ ∈ R. This generalizestheorems of Kaufman [29] and Bluhm [5] and yields new explicit examples of Salemsets. I also prove a multi-dimensional analog of this result. I give applications ofthese results to metrical Diophantine approximation and determine the Hausdorffdimension of E(Q,Ψ, θ) in new cases.iiPrefaceThe main result in Chapter 2 of this thesis is a stronger version of the main result inthe paper [23], published by Izabella Laba and myself. The proof of the main resultin Chapter 2 is a minor reformulation of the proof of the main result in [23].Chapter 3 of this thesis is adapted from my paper [22], which has been submittedfor publication. Some of the text of [22] has been used verbatim in this thesis. I amthe sole author of [22].iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Notation, Conventions, and Basic Definitions . . . . . . . . . . . . . . 41.2 Hausdorff and Fourier Dimension; Salem Sets . . . . . . . . . . . . . 71.3 Restriction Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 The Sharpness of the Exponents in the Mockenhaupt-Mitsis-Bak-Seeger Restriction Theorem . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Theorem 2.0.3 Implies Theorem 2.0.2 . . . . . . . . . . . . . . . . . . 202.2 The Numbers r, s, t, n, Sj, Tj, Nj, and Lj . . . . . . . . . . . . . . . 222.3 The Sequences of Sets (Aj)∞j=0 and (Pj)∞j=0 . . . . . . . . . . . . . . . 232.4 The Measures µ and µj; The Functions fl . . . . . . . . . . . . . . . 242.5 Existence of the Weak Limit . . . . . . . . . . . . . . . . . . . . . . . 252.6 The Ball Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7 The Crucial Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 26iv2.8 Proof of the Crucial Inequality. I. . . . . . . . . . . . . . . . . . . . . 282.9 Proof of the Crucial Inequality. II. . . . . . . . . . . . . . . . . . . . 322.10 Proof of the Crucial Inequality. III. . . . . . . . . . . . . . . . . . . . 342.11 The Fourier Decay of µ . . . . . . . . . . . . . . . . . . . . . . . . . . 362.12 The Convergence of ‖f̂lµj‖2r2r to ‖f̂lµ‖2r2r . . . . . . . . . . . . . . . . . 372.13 A Lower Bound on ‖f̂lµj‖2r2r . . . . . . . . . . . . . . . . . . . . . . . 382.14 The Divergence of ‖f̂lµ‖p/‖fl‖L2(µ) . . . . . . . . . . . . . . . . . . . 423 Explicit Salem Sets and Applications to Metrical Diophantine Ap-proximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1 Remarks on the Proof of Theorems 3.0.1 and 3.0.2 . . . . . . . . . . . 463.2 Motivation: Explicit Salem Sets . . . . . . . . . . . . . . . . . . . . . 473.3 Motivation: Metrical Diophantine Approximation . . . . . . . . . . . 503.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5 Proof of Theorems 3.0.1 and 3.0.2: The Key Function . . . . . . . . . 593.6 Proof of Theorems 3.0.1 and 3.0.2: The Main Lemma . . . . . . . . . 623.7 Proof of Theorems 3.0.1 and 3.0.2: The Measure . . . . . . . . . . . . 653.8 Proof of Lemma 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . 71Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Appendix A Hoeffding’s Inequality . . . . . . . . . . . . . . . . . . . . . 79Appendix B A Fourier Decay Lemma . . . . . . . . . . . . . . . . . . . 81vAcknowledgementsI thank my supervisor, Izabella Laba, for her invaluable guidance throughout myPh.D. She taught me how to be a mathematician.I thank my supervisory committee members, Malabika Pramanik and Akos Mag-yar, for their advice and their time.I thank my fiance´e, Yumi Kondo, for her emotional support throughout my Ph.D.Every mathematical breakthrough I have had during my Ph.D. was made sitting byher side.viDedicationTo YumiviiChapter 1IntroductionWe will begin this chapter by stating the main results of this thesis so that expertscan quickly discern its contribution. The notation, conventions, and basic definitionsused in this thesis are described in Section 1.1. Section 1.2 discusses the conceptsof Hausdorff dimension, Fourier dimension, and Salem sets and describes the con-tribution of this thesis to the study of these concepts. Section 1.3 provides detailedbackground and motivation for the restriction problem. It also explains the contri-bution of this thesis to the study of the restriction problem.This thesis consists of two main parts.The first part concerns the following L2 Fourier restriction theorem.Theorem 1.0.1. Let µ be a compactly supported probability measure on Rd. Supposethere are α, β ∈ (0, d) such thatµ(B(x, r)) . rα ∀x ∈ Rd, r > 0,(A)|µ̂(ξ)| . (1 + |ξ|)−β/2 ∀ξ ∈ Rd.(B)Then for all p ≥ 2(2d− 2α + β)/β we have‖f̂µ‖p . ‖f‖L2(µ) ∀f ∈ L2(µ).(1.0.1)Mockenhaupt [37] and Mitsis [35] independently proved Theorem 1.0.1 for p >12(2d − 2α + β)/β. The endpoint case p = 2(2d − 2α + β)/β was proved by Bakand Seeger [1]. Theorem 1.0.1 generalizes the Tomas-Stein theorem, which is thefollowing.Theorem 1.0.2 (Tomas-Stein theorem). Let µ be the uniform probability measureon the unit sphere Sd−1 in Rd, d ≥ 2. Then for all p ≥ (2d+ 2)/(d− 1) we have‖f̂µ‖p . ‖f‖L2(µ) ∀f ∈ L2(µ).Theorem 1.0.1 implies the Tomas-Stein theorem because (A) and (B) hold withα = β = d− 1 when µ is the uniform probability measure on Sd−1.The range of p in the Tomas-Stein theorem is optimal in the sense that thetheorem does not hold if (2d + 2)/(d − 1) is replaced by any smaller number. Thecounterexample that shows this is called the Knapp example. In the Knapp example,f is taken to be the characteristic function of a small (hence almost flat) sphericalcap (cf. [49, Chapter 7]).Since Theorem 1.0.1 contains the Tomas-Stein theorem as a special case, therange of p in Theorem 1.0.1 is also optimal. However, one can still ask whetherthe range of p in Theorem 1.0.1 is optimal for specific classes of measures. TheTomas-Stein theorem and analogs for curved hypersurfaces besides the sphere wereknown for a long time (cf. [43]). The point of Theorem 1.0.1 is that it also applies tomeasures on fractal sets and to measures on the real line R. Therefore, it is naturalto ask whether the range of p in Theorem 1.0.1 is optimal for such measures. Thisquestion is difficult because there is no obvious analog of the Knapp example (withits almost flat spherical cap) for such measures.The main result of the first part of this thesis is the following theorem, whichshows that the range of p in Theorem 1.0.1 is optimal for measures µ on R thatsatisfy (A) and (B) with β ≤ α.Theorem 1.0.3. Given any 0 < β ≤ α < 1 and given any p∗ < 2(2 − 2α + β)/βthere is a compactly supported probability measure on R that satisfies (A) and (B)but fails to satisfy (2.0.1) whenever p ≤ p∗.2Chapter 2 of this thesis is devoted to Theorem 1.0.3 (and a slightly strongervariant). Theorem 1.0.3 strengthens a result proved by Izabella Laba and myself in2013 [23]. The proof of Theorem 1.0.3 is a minor modification of the proof givenin [23], but it gives a stronger result. More details on how the argument in thisthesis differs from the argument in [23] are also given in Chapter 2. Background onrestriction theorems and further motivation for Theorem 1.0.3 is given in Section 1.3of the current chapter.The second part of this thesis concerns the explicit construction of Salem sets.A Salem set is a subset of Rd whose Hausdorff and Fourier dimension are equal.Every countable set in Rd is Salem set of dimension 0, and Rd itself is a Salem setof dimension d. The classic example of a non-trivial Salem set is a sphere in Rd,which is a Salem set of dimension d − 1. The existence of Salem sets in R of arbi-trary dimension α ∈ (0, 1) was proved by Salem [41] using a random Cantor-typeconstruction. Kahane [27] showed that for every α ∈ (0, d) there is a Salem set inRd of dimension α by considering the images of compact subsets of R under certainstochastic processes (see also Chapters 17 and 18 of [28]). Moreover, Kahane [28,Chapter 18] showed that the image of any compact subset of Rd under fractionalBrownian motion is almost surely a Salem set, indicating that Salem sets are ubiqui-tous amongst random sets. Recently, other random constructions of Salem sets havebeen given by Bluhm [6] and Laba and Pramanik [31]. These random constructionscannot produce any explicit examples of Salem sets. Kaufman [29] was the first tofind an explicit Salem set of dimension α /∈ {0, d− 1, d}. The set Kaufman provedto be Salem is{x ∈ R : ‖qx‖ ≤ |q|−τ for infinitely many q ∈ Z},where τ > 1 and ‖z‖ denotes the distance from z ∈ R to the nearest integer. Thatthis set has Hausdorff dimension 2/(1+τ) is the classical Jarn´ık-Besicovitch theorem[26], [3]. Kaufman showed the Fourier dimension of this set is also 2/(1 + τ). In his3thesis, Bluhm [5] generalized Kaufman’s result by showing that the set{x ∈ R : ‖qx‖ ≤ ψ(|q|) for infinitely many q ∈ Z}is a Salem set for any decreasing function ψ : N → (0,∞). Using a theorem ofGatesoupe [19], explicit Salem sets in Rd of any dimension α between d − 1 and dcan be obtained by considering radial versions of the sets of Kaufman and Bluhm.Besides those already mentioned, no other explicit constructions of Salem setsin R of dimension α 6= 0, 1 or in Rd of dimension α 6= 0, d − 1, d were known untilnow. Moreover, even now, no explicit sets in Rd (Salem or otherwise) with Fourierdimension strictly between 1 and d− 1 are known.The main results of the second part of this thesis generalize the theorems ofKaufman [29] and Bluhm [5] and exhibit many new explicit Salem sets in R. To bemore specific, let Q be an infinite subset of Z, let Ψ : Z→ [0,∞), and let θ ∈ R. Weprove a lower bound on the Fourier dimension of the set{x ∈ R : ‖qx− θ‖ ≤ Ψ(q) for infinitely many q ∈ Q}and prove that it is Salem set for many sets Q and functions Ψ (regardless of thevalue of θ). We also prove a multi-dimensional analog of this result and exhibit newexplicit sets in Rd with Fourier dimension larger than 1. We give applications of theseresults to metrical Diophantine approximation. Chapter 3 of this thesis is devotedto these results. A paper on these results was submitted for publication in 2014 (see[22]).1.1 Notation, Conventions, and Basic DefinitionsIn this section, we explain the notation and conventions we use that may be lessthan universal. We also give some basic definitions.The expression X . Y stands for “there is a constant C > 0 such that X ≤ CY .”The expression X & Y is analogous.4If A ⊆ Rd is a finite set, |A| is the cardinality of A. If A ⊆ Rd is an infinite set,|A| is the Lebesgue outer measure of A.For any A,B ⊆ R and c ∈ R we defineA+B = {a+ b : a ∈ A, b ∈ B} ,c+ A = {c+ a : a ∈ A} ,cA = {ca : a ∈ A} .The diameter of a set A ⊆ Rd is diam(A) = sup {|x− y| : x, y ∈ A}. For x ∈ Rdand r > 0, B(x, r) denotes the open ball in the Euclidean norm with center x andradius r.We let N = {1, 2, . . .} be the set of positive integers and N0 = {0, 1, 2, . . .} be theset of non-negative integers. For n ∈ N, let [n] = {0, . . . , n− 1}.For x ∈ R, ‖x‖ = mink∈Z |x− k| is the distance from x to the nearest integer.The support of a function f : Rd → C is the closure of the set of points x ∈ Rdwhere f(x) 6= 0; it is denoted supp(f).For k ∈ N, we let Ck(Rd) denote the set of functions f : Rd → C that are k-timescontinuously differentiable. We let C∞(Rd) denote the set of functions f : Rd → Cthat are infinitely differentiable. For k ∈ N ∪ {∞}, we let Ckc (Rd) denote the setof functions f ∈ Ck(Rd) with compact support. When no confusion is possible, wewrite Ck for Ck(Rd) and Ckc for Ckc (Rd).Every measure on Rd we consider will be assumed to be defined on a σ-algebraon Rd that contains the Borel σ-algebra on Rd.The support of a measure on Rd is the smallest closed set whose complement hasmeasure 0. Thus any measurable set disjoint from the support has measure 0. Anyset that contains the support of a measure is said to support that measure. We writesupp(µ) for the support of a measure µ on Rd.A probability measure on Rd is a measure µ with µ(Rd) = 1.5Let µ be a measure on Rd. For any µ-measurable function f : Rd → C,‖f‖Lp(µ) =(∫Rd|f(x)|pdµ(x))1/pif 1 ≤ p <∞,‖f‖L∞(µ) = inf{C ≥ 0 : |f(x)| ≤ C for µ-almost every x ∈ Rd}.For p ∈ [1,∞], the space of µ-measurable functions f : Rd → C with ‖f‖Lp(µ) < ∞is denoted Lp(µ). It is a complete normed vector space with norm ‖ ‖Lp(µ) (providedwe identify functions that are equal µ-almost everywhere).The Lebesgue measure on Rd will be denoted by λ. We will often use the notationLp(Rd) = Lp(λ) and ‖f‖p = ‖f‖Lp(Rd) = ‖f‖Lp(λ).The Fourier transform of an L1(λ) function f : Rd → C is defined byf̂(ξ) =∫Rde−2piix·ξf(x)dx for all ξ ∈ Rd.If µ is a measure on Rd and f : Rd → C is an L1(µ) function, we definef̂µ(ξ) =∫Rde−2piix·ξf(x)dµ(x) for all ξ ∈ Rd.(1.1.1)The Fourier transform of a finite complex-valued measure µ on Rd is defined byµ̂(ξ) =∫Rde−2piix·ξdµ(x) for all ξ ∈ Rd.Let V and W be vector spaces with norms ‖ ‖V and ‖ ‖W , respectively. A linearoperator T : V → W is called bounded if there is a constant C > 0 such that‖T (v)‖W ≤ C‖v‖V ∀v ∈ V.A linear operator T : V → W is continuous if and only if it is bounded.61.2 Hausdorff and Fourier Dimension; Salem SetsFor α ≥ 0, the α-dimensional Hausdorff content of a set A ⊆ Rd isHα(A) = infB∑B∈B(diam(B))α,where the infimum is over all countable collections B of balls such that A ⊆⋃B∈B B,and diam(C) denotes the diameter of a set C ⊆ Rd.The Hausdorff dimension of a set A ⊆ Rd is the supremum of all α ∈ [0, d] suchthat Hα(A) > 0. It is denoted by dimH(A).The following lemma is due to Frostman [18] (cf. [34, Chapter 12], [49, Chapter8]).Lemma 1.2.1 (Frostman’s lemma). If A ⊆ Rd is a Borel set, the Hausdorff dimen-sion of A is the supremum of all α ∈ [0, d] such thatµ(B(x, r)) . rα ∀x ∈ Rd, r > 0for some probability measure µ on Rd with supp(µ) ⊆ A.The Fourier dimension of a set A ⊆ Rd is the supremum of all β ∈ [0, d] such that|µ̂(ξ)| . (1 + |ξ|)−β/2 ∀ξ ∈ Rdfor some probability measure µ on Rd with supp(µ) ⊆ A. It is denoted by dimF (A).It is a well-known consequence of Frostman’s lemma thatdimF (A) ≤ dimH(A)(1.2.1)for all Borel sets A ⊆ Rd (cf. [34, Chapter 12], [49, Chapter 8]).A set A ⊆ Rd with dimF (A) = dimH(A) is called a Salem set.Here are some examples of Salem sets. Every countable set in Rd is a Salem setof dimension 0. More generally, every Borel set in Rd of Hausdorff dimension 0 is7a Salem set of dimension 0. The set Rd itself is a Salem set of dimension d. Moregenerally, every set in Rd that contains a ball is a Salem set of dimension d. Everysphere in Rd is a Salem set of dimension d− 1.Here are some examples of non-Salem sets. All k-dimensional planes in Rd withk < d have Hausdorff dimension k and Fourier dimension 0. The middle-thirds Can-tor set in R has Hausdorff dimension (ln 2)/(ln 3) and Fourier dimension 0. Ko¨rner[30] showed that for any given 0 ≤ β < α ≤ 1 there is a compact subset of R withFourier dimension β and Hausdorff dimension α.Note that the concepts of Fourier dimension and Salem sets depend on the am-bient space, unlike the concept of Hausdorff dimension. For example, the real lineR is a Salem set of dimension 1, but a line in R2 (which we can regard as a copyof R embedded in R2) is a non-Salem set with Hausdorff dimension 1 and Fourierdimension 0.The existence of Salem sets in R for every dimension α ∈ (0, 1) was first provedby Salem [41] using a random Cantor-type construction. The existence of Salem setsin Rd for every dimension α ∈ (0, d) was first proved by Kahane [27] (cf. Chapters17 and 18 of [28])) who considered the images of compact subsets of R under certainstochastic processes. Other random constructions of Salem sets have been given byBluhm [6] and Laba and Pramanik [31]. These random constructions do not produceany explicit examples of Salem sets. In fact, it is quite difficult to find explicit Salemsets in Rd of dimensions other than 0, d − 1, and d. The first person to do so wasKaufman [29]. He proved that the set{x ∈ R : ‖qx‖ ≤ |q|−τ for infinitely many q ∈ Z},where τ > 1, is a Salem set of dimension 2/(1 + τ). Bluhm [5] showed that replacing|q|−τ by ψ(|q|) for any decreasing function ψ : N → (0,∞) also yields a Salem set.No other explicit examples of Salem sets in R with dimension 0 < α < 1 were knownuntil now. A theorem of Gatesoupe [19] shows that for any Salem set E contained in[0, 1] of dimension 0 < α < 1, the set of points x ∈ Rd such that |x| ∈ E is a Salemset of dimension d− 1 + α. As it is not difficult to modify Kaufman’s proof to show8that{x ∈ [0, 1] : ‖qx‖ ≤ |q|−τ for infinitely many q ∈ Z}is a Salem set of dimension 2/(1 + τ) when τ > 1, Gatesoupe’s theorem leads toexplicit Salem sets in Rd of every dimension α ∈ (d − 1, d). No explicit examplesof Salem sets in Rd of dimension less than d − 1 are known. Moreover, no explicitsets (Salem or otherwise) in Rd of Fourier dimension strictly between 1 and d−1 areknown.In this thesis we generalize the theorems of Kaufman [29] and Bluhm [5]. To bemore specific, let Q be an infinite subset of Z, let Ψ : Z→ [0,∞), and let θ ∈ R. Weprove a lower bound on the Fourier dimension of the set{x ∈ R : ‖qx− θ‖ ≤ Ψ(q) for infinitely many q ∈ Q} ,and prove that it is Salem for many sets Q and functions Ψ (regardless of the value ofθ). We also prove a multi-dimensional analog of this result and exhibit new explicitsets in Rd with Fourier dimension larger than 1. We give applications of these resultsto metrical Diophantine approximation.1.3 Restriction TheoremsThe restriction problem is one of the most important problems in harmonic analysis.It is connected to many other important problems in mathematical analysis, includ-ing the Kakeya conjecture, the Bochner-Riesz conjecture, the estimation of solutionsto the wave, Schroedinger, and Helmholtz equations, and the local smoothing conjec-ture for partial differential equations (cf. [45]). It is even connected to Montgomery’sconjecture in analytic number theory on the size of Dirichlet sums (see [9] and [10]).The restriction problem was originally proposed by Stein in the late 1960s. Itoriginates from the study of the boundedness of the Fourier transform as a linearoperator between Lp spaces. To motivate it, we begin with the simplest case. Recall9that the Fourier transform of a function f ∈ L1(λ) is defined byf̂(ξ) =∫Rde−2piix·ξf(x)dx for all ξ ∈ Rd.(1.3.1)It is easy to see that the Fourier transform operator f 7→ f̂ is linear. Moreover, theinequality |∫Rd e−2piix·ξf(x)dx| ≤∫Rd |f(x)|dx implies‖f̂‖L∞(µ) ≤ ‖f‖1for any measure µ, meaning the Fourier transform is always a bounded linear operatorfrom L1(λ) to L∞(µ).The restriction problem asks: For which measures µ on Rd and for which valuesof p and q is the Fourier transform a bounded linear operator from Lp(λ) to Lq(µ)?In other words, it asks: For which measures µ on Rd and for which values of p andq do we have an inequality of the form‖f̂‖Lq(µ) ≤ Cp,q‖f‖p(1.3.2)for all functions f in Lp(λ)? Here Cp,q is a constant. The inequality (1.3.2) is calleda restriction inequality.Before saying anything more, there is a technical issue in (1.3.2) we need toaddress. We haven’t defined f̂ for functions f outside of L1(λ). In fact, it is nottrue that the integral in (1.3.1) exists for all f in Lp(λ) when p > 1. However, theintegral in (1.3.1) is certainly defined when f ∈ C∞c (Rd). A standard applicationof Khinchin’s inequality (cf. [49, Chapter 4]) shows that (1.3.2) cannot hold for allf ∈ C∞c (Rd) when p > 2, so we can restrict attention to p ≤ 2. If for some p and q wehave (1.3.2) for all f ∈ C∞c (Rd), then (since C∞c (Rd) is dense in Lp(λ)) the Fouriertransform can be extended uniquely to all f ∈ Lp(λ) in such a way that (1.3.2) holdsfor all f ∈ Lp(λ). Moreover, the extension agrees with (1.3.1) when f ∈ L1(λ). Inlight of this discussion, it is technically cleaner to view the restriction problem asasking: When does an inequality of the form (1.3.2) hold for all f ∈ C∞c (Rd)?10One can view the restriction problem as the problem of generalizing the classi-cal Hausdorff-Young theorem, which says that the Fourier transform is a boundedoperator from Lp(λ) to Lq(λ) if and only if 1 ≤ p ≤ 2 and q = p′ = p/(p− 1).To understand the term restriction in this context, note ‖f̂‖Lq(µ) = ‖f̂ 1S‖Lq(µ),where S is the support of the measure µ. So (1.3.2) says the Fourier restrictionoperator that maps f to f̂ 1S (i.e., maps f to the restriction of the Fourier transformof f to the set S) is a bounded operator from Lp(λ) to Lq(µ).As noted above, the Fourier transform is a bounded operator from L1(λ) to L∞(µ)for any measure µ. If µ is absolutely continuous with respect to Lebesgue measure,the proof of the Hausdorff-Young theorem can be modified to give a restrictioninequality (1.3.2) with p and q depending on the regularity and support of the densityfunction of µ.The restriction problem gets interesting if µ is singular with respect to Lebesguemeasure, so that the support of µ has Lebesgue measure 0. If p > 1, the Fouriertransform of an Lp(λ) function may be infinite on a set of Lebesgue measure 0. If theintersection of that set with the support of µ has positive µ-measure, the left-handside of (1.3.2) will be infinite. Hence f 7→ f̂ may not be a bounded operator fromLp(λ) to Lq(µ) for any q when µ is singular.So which singular measures should we consider? Well, if the support S of µ iscontained in a hyperplane in Rd, we can construct a function that is in Lp(λ) forevery p > 1 and whose Fourier transform is infinite on all of S. For instance, ifψ : Rd−1 → C is any continuous compactly supported function, then the function fdefined byf(x1, . . . , xd) =ψ(x2, . . . , xd)1 + |x1|is in Lp(λ) for every p > 1, but its Fourier transform is infinite everywhere on thehyperplane{ξ ∈ Rd : ξ1 = 0}. Hence we must consider measures µ supported on setsthat are not flat.Perhaps the most natural singular measure to consider is the uniform probabilitymeasure on Sd−1, the unit sphere in Rd. Indeed, this is the measure Stein consideredwhen he first proposed the restriction problem in the late 1960s.11To discuss specific results, it will be convenient to restate the restriction problemin its equivalent dual form. We do so via the following lemma.Lemma 1.3.1. Let µ be a finite measure on Rd. Let p ∈ [1,∞), p′ ∈ (1,∞], andq, q′ ∈ [1,∞] be such that 1p +1p′ = 1,1q +1q′ = 1. The following statements areequivalent.‖ĝ‖Lq(µ) ≤ C‖g‖p ∀g ∈ C∞c (Rd).(1.3.3)‖ĥµ‖p′ ≤ C‖h‖Lq′ (µ) ∀h ∈ Lq′(µ).(1.3.4)Proof. We show only that (1.3.3) implies (1.3.4) as the reverse implication is analo-gous. By Fubini’s theorem, Ho¨lder’s inequality, and (1.3.3), we have∣∣∣∣∫Rdg(y)ĥµ(y)dy∣∣∣∣ =∣∣∣∣∫Rdĝ(x)h(x)dµ(x)∣∣∣∣ ≤ ‖ĝ‖Lq(µ)‖h‖Lq′ (µ) ≤ C‖g‖p‖h‖Lq′ (µ)for all g ∈ C∞c (Rd) and h ∈ Lq′(µ). By applying the formula‖ĥµ‖p′ = sup{∣∣∣∣∫Rdg(y)ĥµ(y)dy∣∣∣∣ : g ∈ C∞c (Rd), ‖g‖p = 1}(cf. [17, Chapter 6]), we obtain (1.3.4).Since µ is assumed to be finite in Lemma 1.3.1, Lq′(µ) ⊆ L1(µ) and ĥµ is well-defined by (1.1.1) for every h ∈ Lq′(µ). An analog of Lemma 1.3.1 can be provedwithout the hypothesis that µ is finite. However, to avoid complications, we consideronly finite measures from now on.In light of Lemma 1.3.1, it is clear an equivalent formulation of the restrictionproblem is the following. For which measures µ on Rd and for which values of p andq do we have an inequality of the form‖f̂µ‖p ≤ Cp,q‖f‖Lq(µ)(1.3.5)for all functions f in Lq(µ)?12Now we discuss some specific results on the restriction problem.We start with a conjecture of Stein.Conjecture 1.3.2 (Stein’s restriction conjecture). If µ is the uniform probabilitymeasure on the unit sphere Sd−1 in Rd, then (1.3.5) holds for all p > 2d/(d − 1)when q =∞.In 1970, Fefferman and Stein (see [16], [43]) proved the conjecture for d = 2, butit remains open for d ≥ 3 (and is nonsense for d = 1).Another foundational result on the restriction problem is the Tomas-Stein theo-rem of 1975 (see [43], [46], [47]).Theorem 1.3.3 (Tomas-Stein theorem). If µ is the uniform probability measure onthe unit sphere Sd−1 in Rd, then (1.3.5) holds for all p ≥ (2d+2)/(d−1) when q = 2.Many researchers have studied the restriction problem for hypersurfaces besidesthe sphere, the restriction problem for surfaces of lower dimension, and analogs ofthe restriction problem in finite fields and in the primes (cf. [2], [12], [20], [36], [38],[39], [45]).Recently, there has been active research into the restriction problem for fractals,where µ is a natural measure on a fractal set.In 2000-2002, Mockenhaupt and Mitsis independently generalized the Tomas-Stein restriction theorem to the fractal setting (see [35], [37]) with the followingtheorem.Theorem 1.3.4. Let µ be a compactly supported probability measure on Rd. Supposethere are α, β ∈ (0, d) such thatµ(B(x, r)) . rα ∀x ∈ Rd, r > 0,(A)|µ̂(ξ)| . (1 + |ξ|)−β/2 ∀ξ ∈ Rd.(B)Then (1.3.5) holds whenever p ≥ 2(2d− 2α + β)/β and q = 2.Note the endpoint case p = 2(2d − 2α + β)/β was not proved by Mockenhauptor Mitsis; it was proved by Bak and Seeger [1] in 2011. Theorem 1.3.4 also holds for13any compactly supported finite measure on Rd, as any finite measure is a constantmultiple of a probability measure. If µ is the uniform probability measure on thesphere in Rd, then (A) and (B) hold with α = β = d− 1, and so (1.3.5) holds for allp ≥ (2d+ 2)/(d− 1) and q = 2. Therefore Theorem 1.3.4 contains the Tomas-Steintheorem as a special case.We now explain how Theorem 1.3.4 is connected to fractals.There is no consensus in the literature on the definition of a fractal, but we willnot need a formal definition anyway. Informally, one may think of a fractal as a setin Rd that exhibits self-similarity, structure at arbitrarily small scales, and local andglobal irregularity.The complicated structure of fractal sets is often summarized by various numericaldimensions, such as Hausdorff dimension and Fourier dimension. For example, themiddle-thirds Cantor set (one of the most famous fractals) has Hausdorff dimension(ln 2)/(ln 3) and Fourier dimension 0.Condition (A) is closely connected to Hausdorff dimension. Indeed, recall thatLemma 1.2.1 says the Hausdorff dimension of a Borel set A ⊆ Rd is the supremumof all α ∈ [0, d] for which there exists a probability measure µ supported on A suchthat condition (A) holds. Consequently, if µ is a probability measure satisfying (A)for some α ∈ [0, d], then α ≤ dimH(supp(µ)).Condition (B) concerns the Fourier decay rate of µ and is connected to Fourierdimension. The original proofs of the Tomas-Stein theorem for the sphere and analogsof it for other surfaces hinge on deducing an estimate of the Fourier decay rateof the relevant measure from the curvature of the surface on which the measureis supported. By the definition of Fourier dimension in Section 1.2, a probabilitymeasure that satisfies condition (B) must satisfy β ≤ dimF (supp(µ)).Complimentary to the Tomas-Stein theorem is the important fact that the rangeof p in the theorem cannot be enlarged. This is shown by the classic example ofKnapp, in which one takes f to be the characteristic function of a small (hencealmost flat) spherical cap (cf. [49, Chapter 7]). As Theorem 1.3.4 contains theTomas-Stein theorem as a special case, the range of p in Theorem 1.3.4 also cannotbe enlarged in general. However, the point of Theorem 1.3.4 is that it also applies to14measures supported on fractal sets and to measures on R. Therefore, it is interestingto ask whether the range of p in Theorem 1.3.4 is best possible for such measures.In 2013, Izabella Laba and I addressed this question (see [23]).To understand the result of Laba and myself in [23], first observe that for aprobability measure µ on R that satisfies (A) for some α ∈ (0, 1) and satisfies (B)for every β < α, the inequality (1.3.5) holds whenever p > 2(2 − α)/α and q = 2.The result Laba and I established in [23] implies the following.Theorem 1.3.5. Given any 0 < α < 1 and given any p∗ < 2(2 − α)/α there is acompactly supported probability measure on R that satisfies (A) and satisfies (B) forevery β < α but does not satisfy (1.3.5) when p ≤ p∗ and q = 2.The construction in [23] of the measure µ is based on the randomized Cantor-type construction of Salem sets by Laba and Pramanik [31]. Laba and I modified theconstruction so that the Cantor set that supports µ contains a smaller deterministicCantor set. Care must be taken in including the deterministic Cantor set to avoiddestroying the Fourier decay estimates on µ. The inequality (1.3.5) is shown to beviolated when p ≤ p∗ and q = 2 for any prescribed constant Cp,q by taking f to bethe characteristic function of a finite iteration of the deterministic Cantor set. Theargument ultimately hinges on counting solutions to certain linear equations in theendpoints of the finite iterations of the larger Cantor set. To ensure the number ofsolutions is sufficiently large, the embedded deterministic Cantor set is constructedwith the endpoints of each finite iteration in a generalized arithmetic progression.In this thesis, we improve the result of [23]. In particular, we prove the following.Theorem 1.3.6. Given any 0 < β ≤ α < 1 and given any p∗ < 2(2 − 2α + β)/βthere is a compactly supported probability measure on R that satisfies (A) and (B)but does not satisfy (1.3.5) when p ≤ p∗ and q = 2.This is a restatement of Theorem 1.0.3. In Chapter 2, we discuss the minordifferences between the proof of Theorem 1.3.6 (which we give in this thesis) and theproof of the result of [23].Theorem 1.3.6 establishes the sharpness of range of p in Theorem 1.3.4 for theclass of probability measures µ on R that satisfy (A) and (B) for some 0 < β ≤ α < 1.15In particular, Theorem 1.3.6 covers those measures that satisfy (A) and (B) withα = β. Theorem 1.3.5 does not cover such measures; it only covers those measuresthat satisfy (A) for some 0 < α < 1 and satisfy (B) for β arbitrarily close to (butstrictly less than) α. Theorem 1.3.6 establishes sharpness for those measures thatsatisfy 0 < β < α < 1. Theorem 1.3.5 does not cover such measures because itonly permits taking p∗ up to the number 2(2 − α)/α, not up to the larger number2(2− 2α + β)/β.Chen [11] modified the argument used by Laba and myself in [23] to eliminatethe dependence of the measure on p∗. In particular, Chen [11] provedTheorem 1.3.7. Given any 0 < β ≤ α < 1 there is a compactly supported probabilitymeasure on R that satisfies (A) and (B) but does not satisfy (1.3.5) when p < 2(2−2α + β)/β and q = 2.Evidently, Theorem 1.3.7 implies Theorem 1.3.6. Theorem 1.3.7 is perhaps alsoa more satisfying sharpness statement than Theorem 1.3.6. The modification of theargument of [23] that we present in this thesis to prove Theorem 1.3.6 does not followthe modification of [11]. However, we do borrow some of his notation.16Chapter 2The Sharpness of the Exponents inthe Mockenhaupt-Mitsis-Bak-Seeger Restriction TheoremIn this chapter, we study the sharpness of the range of exponents in the followinggeneral L2 restriction theorem of Mockenhaupt [37] and Mitsis [35] (with the endpointdue to Bak and Seeger [1]).Theorem 2.0.1. Let µ be a compactly supported probability measure on Rd. Supposethere are α, β ∈ (0, d) such thatµ(B(x, r)) . rα ∀x ∈ Rd, r > 0,(A)|µ̂(ξ)| . (1 + |ξ|)−β/2 ∀ξ ∈ Rd.(B)Then for all p ≥ 2(2d− 2α + β)/β we have‖f̂µ‖p . ‖f‖L2(µ) ∀f ∈ L2(µ).(2.0.1)We will consider the following question. Is the range of p in Theorem 2.0.1 sharp?In other words, does the theorem still hold if 2(2d−2α+β)/β is replaced by a smallernumber p∗?17This question, as stated, is too general. The Knapp example, where f is thecharacteristic function of a small spherical cap and µ is the uniform probabilitymeasure on the unit sphere in Rd, shows the range of p is sharp (cf. [49, Chapter 7]).However, we can consider the sharpness for specific sets of measures. The noveltyof Theorem 2.0.1 compared to earlier restriction theorems is that it also applies tomeasures on fractal sets and to measures on R. Therefore, it is natural to considerthe sharpness of the range of p in Theorem 2.0.1 for such measures.In this chapter, we will show the range of p in Theorem 2.0.1 is sharp for measuresµ on R that satisfy (A) and (B) with β ≤ α. In particular, I will prove the followingtheorem.Theorem 2.0.2. Given any 0 < β ≤ α < 1 and given any p∗ < 2(2 − 2α + β)/βthere is a compactly supported probability measure on R that satisfies (A) and (B)but fails to satisfy (2.0.1) whenever p ≤ p∗.Theorem 2.0.2 will be deduced from the following stronger result.Theorem 2.0.3. Let 0 < β0 ≤ α0 < 1 with α0 = ln a/ ln c and β0 = (2 ln a−ln b)/ ln cfor some a, b, c ∈ N. Given any p∗ < 2(2−2α0+β0)/β0, there is a compactly supportedprobability measure on R that satisfies the following.µ(B(x, r)) . rα0 ∀x ∈ R,∀r > 0.(2.0.2)|µ̂(ξ)| . (1 + |ξ|)−β0/2(ln(1 + |ξ|))1/2 ∀ξ ∈ R.(2.0.3)There is a sequence of functions fl ∈ L2(µ) such thatliml→∞‖f̂lµ‖p‖fl‖L2(µ)=∞ ∀p ≤ p∗.(2.0.4)In section 2.1, we show that Theorem 2.0.3 implies Theorem 2.0.2. The rest ofthe chapter constitutes the proof of Theorem 2.0.3.In 2013, Izabella Laba and I [23] proved (implicitly) the following theorem, whichis the special case of Theorem 2.0.3 where α0 = β0.18Theorem 2.0.4. Let 0 < α0 < 1 with α0 = ln a/ ln c for some a, c ∈ N. Givenany p∗ < 2(2− α0)/α0, there is a compactly supported probability measure on R thatsatisfies the following.µ(B(x, r)) . rα0 ∀x ∈ R, ∀r > 0.(2.0.5)|µ̂(ξ)| . (1 + |ξ|)−α0/2(ln(1 + |ξ|))1/2 ∀ξ ∈ R.(2.0.6)There is a sequence of functions fl ∈ L2(µ) such thatliml→∞‖f̂lµ‖p‖fl‖L2(µ)=∞ ∀p ≤ p∗.(2.0.7)Theorem 2.0.4 can be deduced from Theorem 2.0.3 by taking a = b.Just as Theorem 2.0.3 implies Theorem 2.0.2, Theorem 2.0.4 implies the followingtheorem, which says the the range of p in Theorem 2.0.1 is sharp for measures µ onR that satisfy (A) for some α ∈ (0, 1) and satisfy (B) for every β < α.Theorem 2.0.5. Given any 0 < α < 1 and given any p∗ < 2(2 − α)/α there is acompactly supported probability measure on R that satisfies (A) and satisfies (B) forevery β < α but fails to satisfy (2.0.1) whenever p ≤ p∗.Let us compare what Theorem 2.0.2 and Theorem 2.0.5 tell us.Theorem 2.0.2 establishes the sharpness of range of p in Theorem 2.0.1 for theclass of probability measures µ on R that satisfy (A) and (B) for some 0 < β ≤ α < 1.In particular, it covers those measures that satisfy (A) and (B) with α = β. Theorem2.0.5 does not cover such measures. Theorem 2.0.5 only covers those measures thatsatisfy (A) for some 0 < α < 1 and satisfy (B) for all β < α.Furthermore, Theorem 2.0.5 does not establish the sharpness of the range ofp in Theorem 2.0.1 for the class of probability measures µ on R that satisfy (A)and (B) for some 0 < β < α < 1, while Theorem 2.0.2 does. It is true thatTheorem 2.0.5 produces, for any prescribed 0 < β < α < 1 and p∗ < 2(2 − α)/α, aprobability measure µ that satisfies (A) and (B) but does not satisfy (2.0.1) whenp ≤ p∗. However, establishing sharpness for the class of measures under consideration19requires exhibiting, for any prescribed 0 < β < α < 1 and p∗ < 2(2 − 2α + β)/β, aprobability measure µ that satisfies (A) and (B) but does not satisfy (2.0.1) whenp ≤ p∗. Theorem 2.0.5 does not allow us to take p∗ all the way up to 2(2−2α+β)/β.It only permits taking p∗ up to the number 2(2 − α)/α, which is strictly less than2(2− 2α + β)/β when β < α.The proof of Theorem 2.0.3 is a straightforward modification of the proof ofTheorem 2.0.4 given in [23]. It is based ultimately on the randomized Cantor-typeconstruction of Salem sets by Laba and Pramanik [31]. We modify the constructionso that the Cantor set that supports µ contains a smaller deterministic Cantor set.The functions f` are the characteristic functions of the finite iterations of this de-terministic Cantor set. We must be careful that the deterministic Cantor set is notso large that it destroys the Fourier decay estimates furnished by the randomness ofthe larger Cantor set. The divergence of ‖f̂`µ‖p/‖f`‖L2(µ) relies on counting solutionsto linear equations in the endpoints of the finite iterations of the larger Cantor set.The deterministic Cantor set is chosen with the endpoints of its finite iterations ina multi-scale arithmetic progression so that this count is sufficiently large to ensuredivergence.The difference between the proof of Theorem 2.0.3 and the proof of Theorem 2.0.4given in [23] is the presence of β0 = (2 ln a− ln b)/ ln c in addition to α0 = ln a/ ln c.This is handled by simply using β0 instead of α0 in all estimates related to theFourier decay of µ. The form of β0 is chosen so that the length (s = nα0−β0/2 in theterminology to be introduced in Section 2.2) of the single-scale arithmetic progression(which underlies the multi-scale arithmetic progression) is an integer.2.1 Theorem 2.0.3 Implies Theorem 2.0.2In this section, we show that Theorem 2.0.3 implies Theorem 2.0.2.Let 0 < β ≤ α < 1 and p∗ < 2(2 − 2α + β)/β be given. Choose > 0 small20enough that α + < 1 andp∗ <2(2− 2(α + ) + (β + ))β + .(2.1.1)Choose c ∈ N large enough that1ln c<2Let a ∈ N be the smallest positive integer thatα +2ln aln c−1ln c>ln aln c−2.(2.1.3)Therefore, by combining (2.1.2) and (2.1.3), we obtainα < α +2< α0 < α + ,(2.1.4)whereα0 =ln aln c.Let b ∈ N be the largest positive integer such thatln bln c< 2α0 − β.(2.1.5)Then2α0 − β ≤ln(b+ 1)ln c 1 and2r ≥ p∗. Define n = c2m, t = a2m = nα0 , and s = bm = nα0−β0/2, where m is a fixed22positive integer chosen so large thatp∗ <2(2− 2α0 + β0)β0−4 ln rβ0 lnn.(2.2.1)Define Sj = sj, Tj = tj, Nj = nj, and Lj+1 = ln(400(j + 1)Nj+1) for j ∈ N0.2.3 The Sequences of Sets (Aj)∞j=0 and (Pj)∞j=0In this section, we describe the sequences of sets (Aj)∞j=0 and (Pj)∞j=0.Each Aj will be the set of left endpoints of the intervals in the j-th iteration ofthe Cantor set supporting µ. Each Pj will be the set of left endpoints of the intervalsin the j-th iteration of the smaller deterministic Cantor set we embed within theCantor set supporting µ.Let (Pj)∞j=0 be the sequence of sets defined byP0 = {0} , Pj+1 = Pj +1Nj+1{1, . . . , s} =⋃a∈Pja+1Nj+1{1, . . . , s} .We will construct inductively a sequence of sets (Aj)∞j=0 of the form• A0 = {0}• Aj+1 =⋃a∈Aja+ Aj+1,a• Aj+1,a ⊆1Nj+1[n]• |Aj+1,a| = Tj+1• Aj ⊆ Pj.Note|Aj| = Tj, Aj ⊆1Nj[Nj], Aj,a ⊆1Nj{1, . . . , s}for all j ∈ N. The details of the construction of (Aj)∞j=0 will be given in Section 2.8.232.4 The Measures µ and µj; The Functions flIn this section, we define the probability measures µ and µj (j ∈ N0) on R and thefunctions fl : R→ C (l ∈ N0).For j ∈ N0, defineEj = Aj + [0, N−1j ]and define µj to be the uniform probability measure on Ej, i.e.,dµj =1|Ej|1Ejdx =NjTj∑a∈Aj1[a,a+N−1j ]dx.Define µ to be the weak limit of (µj)∞j=0, i.e.,µ = limj→∞µj.We prove the weak limit exists in Section 2.5. A good reference on weak convergenceis [4]. Notesupp(µ) =∞⋂j=1Ej.Let j ∈ N0 and a ∈ Aj. Thenµj([a, a+N−1j ]) =1Tj,and in factµk([a, a+N−1j ]) =1Tj∀k ∈ N0, k ≥ j.Consequently,µ([a, a+N−1j ]) =1Tj.For l ∈ N0, defineFl = Pl + [0, N−1l ) and fl = 1Fl .242.5 Existence of the Weak LimitIn this section, we prove the existence of the weak limitµ = limj→∞µj.For all other sections in this chapter, Fj is the set defined in Section 2.4. However,in this section, Fj will be the cumulative distribution function of µj, i.e.,Fj(t) = µj((−∞, t]), ∀t ∈ R.Fix j ∈ N0. For a ∈ Aj, we have Fj+1(a) = Fj(a) by the definitions of µj+1 andµj. For t ∈ R, let a(t) be the smallest element of N−1j [Nj] such that a(t) ≤ t. Then|Fj+1(t)− Fj(t)| = |µj+1([a(t), t])− µj([a(t), t])|≤ µj+1([a(t), t]) + µj([a(t), t])≤ µj+1([a(t), a(t) +N−j]) + µj([a(t), a(t) +N−j])≤2Tj.Therefore‖Fj+1 − Fj‖∞ ≤2Tj∀j ∈ N0.Since Tj = njα0 for j ∈ N0, we have∑∞j=0 T−1j < ∞. It follows that (Fj)∞j=1 isa uniformly convergent sequence of continuous cumulative distribution functions.Therefore the limit F is a continuous cumulative distribution function of a probabilitymeasure µ. Hence (µj)∞j=1 converges weakly to µ.252.6 The Ball ConditionIn this section, we will show that µ(|I|) . |I|α0 for all intervals I ⊆ R. This isequivalent to (2.0.2) in Theorem 2.0.3.Let I be an interval in R. If |I| > 1, thenµ(I) ≤ µ(R) = 1 ≤ |I|α0since µ is a probability measure. Now suppose |I| ≤ 1. Choose j ∈ N0 such thatN−1j+1 ≤ |I| ≤ N−1j . Assume I intersects supp(µ) (otherwise µ(I) = 0). Notesupp(µ) =∞⋂j=1Ej =∞⋂j=1⋃a∈Aj[a, a+N−1j ].Then I intersects an interval [a, a + N−1j ] for some a ∈ Aj. There are at most twosuch intervals, say J1 and J2, because |I| ≤ N−1j and Aj ⊆ N−1j [Nj]. Thereforeµ(I) = µ(I ∩ J1) + µ(I ∩ J2) ≤ µ(J1) + µ(J2)=1Tj+1Tj=2nα0Nα0j+1≤ 2nα0|I|α0 .2.7 The Crucial InequalityIn this section, we state the crucial Fourier analytic inequality, explain the steps inits proof, and explain its use.Here is the crucial Fourier analytic inequality:∞∑j=0|f̂lµj+1(k)− f̂lµj(k)| . |k|−β0/2(ln(e+ |k|))1/2(2.7.1)for all k ∈ Z with k 6= 0 and all l ∈ N0.In the next three sections, we prove we can construct the sets (Aj)∞j=0 so that(2.7.1) holds.26First we show we can construct the sets (Aj)∞j=0 so that∣∣∣∣∣∣1Tj∑a∈Aj∩FlYa(k)∣∣∣∣∣∣. N−β0/2j+1 L1/2j+1(2.7.2)for all k ∈ Z and all j, l ∈ N0, j ≥ l. HereYa(k) = e−2piika1t∑b∈Aj+1,ae−2piikb −1n∑b∈[n]/Nj+1e−2piikb .Next we show|f̂lµj+1(k)− f̂lµj(k)| ≤ 2 min{1,Nj+1|k|}∣∣∣∣∣∣1Tj∑a∈Aj∩FlYa(k)∣∣∣∣∣∣(2.7.3)for all k ∈ Z, k 6= 0 and all j, l ∈ N with j ≥ l.Finally we show that∞∑j=0min{1,Nj+1|k|}N−β0/2j+1 L1/2j+1 . |k|−β0/2(ln(e+ |k|))1/2(2.7.4)for all k ∈ Z, k 6= 0.Evidently (2.7.1) follows upon combining (2.7.2), (2.7.3), and (2.7.4).Once (2.7.1) is established, we will use it to prove|µ̂(ξ)| . (1 + |ξ|)−β0/2(ln(e+ |ξ|))1/2 ∀ξ ∈ R.(2.7.5)andlimj→∞‖f̂lµj‖2r2r = ‖f̂lµ‖2r2r ∀l ∈ N0.(2.7.6)Note (2.7.5) is (2.0.3) in Theorem 2.0.3.272.8 Proof of the Crucial Inequality. I.In this section, we will define (Aj)∞j=0 so that (2.7.2) holds. Recall that (2.7.2) is theinequality∣∣∣∣∣∣1Tj∑a∈Aj∩FlYa(k)∣∣∣∣∣∣. N−β0/2j+1 L1/2j+1for all k ∈ Z and all j, l ∈ N0, j ≥ l. Here (as elsewhere)Ya(k) = e−2piika1t∑b∈Aj+1,ae−2piikb −1n∑b∈[n]/Nj+1e−2piikb .We will define (Aj)∞j=0 inductively. Fix j ∈ N0. Assume Aj containing Pj is given.We will choose Aj+1 =⋃a∈Aja+ Aj+1,a by choosing Aj+1, a for each a ∈ Aj.We will need the following version of Hoeffding’s inequality.Lemma 2.8.1. If X1, . . . , Xn are independent complex-valued random variables withEXi = 0 and |Xi| ≤ 1 for i = 1, . . . , n, thenP(∣∣∣∣∣1nn∑i=1Xi∣∣∣∣∣≥ u)≤ 4 exp(−nu2/4)for all u ≥ 0.This version of Hoeffding’s inequality for complex-valued random variables isdeduced from the standard version for real-valued random variables in Appendix A.Let Bj+1 be the collection of all size tj+1 subsets of N−1j+1[n]. There are(nt)setsin Bj+1. Moreover, each fixed b ∈ N−1j+1[n] is contained in exactly(n−1t−1)sets in Bj+1.For each a ∈ Aj, choose a set Bj+1,a independently and uniformly at random28from Bj+1. For each a ∈ Aj and k ∈ Z, define the random variableXa(k) =12e−2piiak1t∑b∈Bj+1,ae−2piikb −1n∑b∈N−1j+1[n]e−2piikb .For each a ∈ Aj, we haveE∑b∈Bj+1,ae−2piiξb =1(nt)∑B∈Bj+1∑b∈Be−2piiξb=1(nt)∑b∈N−1j+1[n](n− 1t− 1)e−2piiξb=tn∑b∈N−1j+1[n]e−2piiξb.It follows that EXa(k) = 0 for all a ∈ Aj and k ∈ Z. Clearly |Xa(k)| ≤ 1 for alla ∈ Aj and k ∈ Z.Fix k ∈ [Nj+1]. The random variables (Xa(k))a∈Aj are independent because ofhow we choose the sets (Bj+1,a)a∈Aj . Note|Aj ∩ Fl| = sltj−l = (Sl/Tl)Tj ∀l ∈ N0, l ≤ j.Define ul,j ≥ 0 byu2l,j = 4(Sl/Tl)−1T−1j ln(400(j + 1)Nj+1) ∀l ∈ N0, l ≤ j.Lemma 2.8.1 impliesP∣∣∣∣∣∣1(Sl/Tl)Tj∑a∈Aj∩FlXa(k)∣∣∣∣∣∣≥ ul,j ≤ 4 exp(−(Sl/Tl)Tju2l,j/4) =1100(j + 1)Nj+1.29ThenP∣∣∣∣∣∣1TjS−1l∑a∈Aj∩FlXa(k)∣∣∣∣∣∣≥ ul,j for some k ∈ [Nj+1], l ∈ {0, . . . , j} ≤1100.Therefore there is some realization of the randomly chosen sets (Bj+1,a)a∈Aj suchthat∣∣∣∣∣∣1(Sl/Tl)Tj∑a∈Aj∩FlXa(k)∣∣∣∣∣∣< ul,j(2.8.1)for all k ∈ [Nj+1] and all l ∈ N0, l ≤ j. Fix such a realization. Since Xa(k) is periodicwith period Nj+1 when a ∈ Aj, the inequality (2.8.1) holds for all k ∈ Z.We cannot simply define Aj+1,a = Bj+1,a because thenAj+1 =⋃a∈Aja+ Aj+1,amay not containPj+1 =⋃a∈Pja+N−1j+1 {1, . . . , s} .We define Aj+1,a for a ∈ Aj as follows. For a /∈ Pj, define Aj+1,a = Bj+1,a. Fora ∈ Pj, define Aj+1,a to be the set obtained by first removing s arbitrary elementsof Bj+1,a and then replacing them by the elements of N−1j+1 {1, . . . , s}. Then Pj+1 ⊆Aj+1.Recall that for a ∈ Aj and k ∈ Z we haveXa(k) = e−2piika1t∑b∈Bj+1,ae−2piikb −1n∑b∈[n]/Nj+1e−2piikb ,Ya(k) = e−2piika1t∑b∈Aj+1,ae−2piikb −1n∑b∈[n]/Nj+1e−2piikb .30For a /∈ Pj, we have Ya(k) = Xa(k). For a ∈ Pj,∑b∈Aj+1,ae−2piikb differs from∑b∈Bj+1,ae−2piikb by at most s terms, hence|Ya(k)−Xa(k)| =1t∣∣∣∣∣∣∑b∈Aj+1,ae−2piikb −∑b∈Bj+1,ae−2piikb∣∣∣∣∣∣≤2st.Therefore, for all l ∈ N0, l ≤ j, we have∣∣∣∣∣∣1Tj∑a∈Aj∩FlYa(k)−1Tj∑a∈Aj∩FlXa(k)∣∣∣∣∣∣≤1Tj∑a∈Aj∩Fl|Ya(k)−Xa(k)|=1Tj∑a∈Pj|Ya(k)−Xa(k)|≤|Pj|Tj2st=2Sj+1Tj+1.Combining this with (2.8.1) yields∣∣∣∣∣∣1Tj∑a∈Aj∩FlYa(k)∣∣∣∣∣∣< (Sl/Tl)1/2(4T−1j ln(400(j + 1)Nj+1))1/2 +2Sj+1Tj+1.Recall that s = nα0−β0/2, t = nα0 , Sj = sj, Tj = tj, Nj = nj, and Lj+1 = ln(400(j +1)Nj+1). It follows that∣∣∣∣∣∣1Tj∑a∈Aj∩FlYa(k)∣∣∣∣∣∣. N−β0/2j+1 L1/2j+1.312.9 Proof of the Crucial Inequality. II.In this section, we prove (2.7.3), i.e.,|f̂lµj+1(k)− f̂lµj(k)| ≤ 2 min{1,Nj+1|k|}∣∣∣∣∣∣1Tj∑a∈Aj∩FlYa(k)∣∣∣∣∣∣for all k ∈ Z, k 6= 0 and all j, l ∈ N with j ≥ l. RecallYa(k) = e−2piika1t∑b∈Aj+1,ae−2piikb −1n∑b∈[n]/Nj+1e−2piikb .Fix j, l ∈ N0 with j ≥ l. For a ∈ Aj and x ∈ R, definega,j(x) = Nj1[a,a+N−1j ](x) = N j1[0,1](Nj(x− a)).Thendµj(x) =1Tj∑a∈Ajga,j(x)dx,and sofldµj(x) =1Tj∑a∈Ajfl(x)ga,j(x)dx =1Tj∑a∈Aj∩Flga,j(x)dx.For ξ ∈ R,ĝa,j(ξ) = e−2piiaξ1̂[0,1](ξ/Nj).Thereforef̂lµj(ξ) =1Tj1̂[0,1](ξ/Nj)∑a∈Aj∩Fle−2piiaξ.(2.9.1)32Likewisef̂lµj+1(ξ) =1Tj1̂[0,1](ξ/Nj+1)∑a∈Aj+1∩Fle−2piiaξ.(2.9.2)We now rewrite (2.9.1) and (2.9.2). First, by the geometric sum formula,−2pii(ξ/Nj)1̂[0,1](ξ/Nj) = e−2piiξ/Nj − 1 = e−2pii(ξ/Nj+1)n − 1= (e−2pii(ξ/Nj+1) − 1)∑k∈[n]e−2pii(ξ/Nj+1)k= −2pii(ξ/Nj+1)1̂[0,1](ξ/Nj+1)∑k∈[n]e−2pii(ξ/Nj+1)n= −2pii(ξ/Nj)1̂[0,1](ξ/Nj+1)1n∑b∈N−1j+1[n]e−2piiξb.Therefore1̂[0,1](ξ/Nj) = 1̂[0,1](ξ/Nj+1)1n∑b∈N−1j+1[n]e−2piiξb.Applying this to (2.9.1) givesf̂lµj(ξ) =1Tj1̂[0,1](ξ/Nj+1)∑a∈Aj∩Fle−2piiaξ1nj+1∑b∈N−1j+1[nj+1]e−2piiξb.(2.9.3)On the other hand, sinceAj+1 =⋃a∈Aja+ Aj+1,a,33we havef̂lµj+1(ξ) =1Tj+11̂[0,1](ξ/Nj+1)∑a∈Aj∩Fl∑b∈Aj+1,ae−2piiξ(a+b)(2.9.4)=1Tj1̂[0,1](ξ/Nj+1)∑a∈Aj∩Fle−2piiξa1t∑b∈Aj+1,ae−2piiξb.Fix k ∈ Z, k 6= 0. Comparing (2.9.3) and (2.9.4) yieldsf̂lµj+1(k)− f̂lµj(k) = 21̂[0,1](k/Nj+1)1Tj∑a∈Aj∩FlYa(k).(2.9.5)Since|1̂[0,1](k/Nj+1)| ≤ 1̂[0,1](0) = 1and|1̂[0,1](k/Nj+1)| =∣∣∣∣e−2piik/Nj+1 − 1−2piik/Nj+1∣∣∣∣ ≤1pi|k|/Nj+1,it follows from (2.9.5) that|f̂lµj+1(k)− f̂lµj(k)| ≤ 2 min{1,Nj+1|k|}∣∣∣∣∣∣1Tj∑a∈Aj∩FlYa(k)∣∣∣∣∣∣.2.10 Proof of the Crucial Inequality. III.In this section, we prove (2.7.4), i.e.,∞∑j=0min{1,Nj+1|k|}N−β0/2j+1 L1/2j+1 . (ln(e+ |k|))1/2|k|−β0/2for all ∀k ∈ Z, k 6= 0.34Fix k ∈ Z, k 6= 0. Let j∗ be the smallest non-negative integer such that Nj∗+1 >|k|. Then Nj∗ ≤ |k|. Our strategy is to the split the sum above into a sum over0 ≤ j < j∗ and a sum over j ≥ j∗ and estimate each separately. For any j ∈ N0,Lj+1 = ln(400(j + 1)Nj+1) ≤ 10 ln(Nj+1).If 0 ≤ j < j∗, thenLj+1 ≤ 10 ln(Nj∗) ≤ 10 ln |k|.If j ≥ j∗, thenLj+1 ≤ 10 ln(Nj−j∗+1Nj∗) ≤ 10 ln(Nj−j∗+1|k|)≤ 20 ln(Nj−j∗+1) ln(e+ |k|) = 20(j − j∗ + 1) ln(n) ln(e+ |k|).For the sum over 0 ≤ j < j∗, we have1|k|∑0≤j 1and sincelimj→∞(f̂lµ(ξ)− f̂lµj(ξ))→ 0pointwise for every ξ ∈ R, the dominated convergence theorem implieslimj→∞‖f̂lµ(ξ)− f̂lµj(ξ)‖2r = 0.It follows by the triangle inequality thatlimj→∞‖f̂lµj‖2r2r = ‖f̂lµ‖2r2r.2.13 A Lower Bound on ‖f̂lµj‖2r2rIn this section, we show that‖f̂lµj‖2r2r ≥ Cr(SlTl)2r Nlrl+1Sl∀j, l ∈ N0, j ≥ l,(2.13.1)where Cr > 0 is a constant depending only on r.Fix j, l ∈ N0 with j ≥ l. Recall from (2.9.1) thatf̂lµj(ξ) =1Tj1̂[0,1](ξ/Nj)∑a∈Aj∩Fle−2piiξa38for all ξ ∈ R. Thereforef̂lµj(ξ) =1Tj1̂[− 12 ,12 ](ξ/Nj)e−piξ/Nj∑a∈Aj∩Fle−2piiξa.Then|f̂lµj(ξ)|2r =1T 2rj∣∣∣∣∣∣∑a∈Aj∩Fle−2piiξa∣∣∣∣∣∣2r∣∣∣1̂[− 12 ,12 ](ξ/Nj)e−piiξ/Nj∣∣∣2r=1T 2rj∣∣∣∣∣∣∑a∈Aj∩Fle−2piiξa∣∣∣∣∣∣2r(1̂[− 12 ,12 ](ξ/Nj))2r=1T 2rj∑a1,...,a2r∈Aj∩Fle−2piiξ(∑rs=1 as−∑2rs=r+1 as)1̂∗2r[− 12 ,12 ](ξ/Nj).Here we have used1̂[− 12 ,12 ](ξ/Nj) =e−piiξ/Nj − epiiξ/Nj−2piiξ/Nj=sin(piξ/Nj)piξ/Nj∈ R,and the formulas |z|2r = zr z r and(f̂)2r= f̂ ∗2r, where f ∗2r is the 2r-fold convolutionof f . Therefore‖flµj‖2r2r =∫R|f̂lµj(ξ)|2rdξ = Nj∫R|f̂lµj(Njξ)|2rdξ=NjT 2rj∑a1,...,a2r∈Aj∩Fl∫Re−2piiξNj(∑rs=1 as−∑2rs=r+1 as)1̂∗2r[− 12 ,12 ](ξ)dξ.By Fourier inversion,‖flµj‖2r2r =NjT 2rj∑a1,...,a2r∈Aj∩Fl1∗2r[− 12 ,12 ](Nj(r∑s=1as −2r∑s=r+1as)).39Since 1∗2r[− 12 ,12 ]≥ 0, all the terms in the sum are non-negative. Therefore‖flµj‖2r2r ≥NjT 2rj1∗2r[− 12 ,12 ](0)Mj,l,r,whereMj,l,r =∣∣∣∣∣{(a1, . . . , a2r) ∈ (Aj ∩ Fl)2r :r∑s=1as =2r∑s=r+1as}∣∣∣∣∣.Take Cr = 1∗2r[− 12 , 12 ](0).To prove (2.13.1) now, it remains to establish the lower boundMj,l,r ≥T 2rjNj(SlTl)2r Nlrl+1Sl.(2.13.2)Define(Aj ∩ Fl)⊕r ={r∑s=1as : a1, . . . , ar ∈ Aj ∩ Fl}.We haveMj,l,r =∑b∈(Aj∩Fl)⊕r(g(b))2,whereg(b) =∣∣∣∣∣{(a1, . . . , ar) ∈ (Aj ∩ Fl)r :r∑s=1as = b}∣∣∣∣∣So by the Cauchy-Schwarz inequality∑b∈(Aj∩Fl)⊕rg(b)2≤ |(Aj ∩ Fl)⊕r|Mj,l,r.(2.13.3)40Clearly∑b∈(Aj∩Fl)⊕rg(b) = |(Aj ∩ Fl)r| =(SlTlTj)r(2.13.4)To estimate |(Aj ∩ Fl)⊕r|, note each a ∈ Aj ∩ Fl has a digit representation of theforma =l∑k=1a(k)Nk+j∑k=l+1a(k)Nk,where a(k) ∈ {1, . . . , s} for k = 1, . . . , l and a(k) ∈ [n] for k = l + 1, . . . , j. Thereforeeach a ∈ Aj ∩ Fl is of the forma =l∑k=1a(k)Nk+a∗Nl,where a(k) ∈ {1, . . . , s} for k = 1, . . . , l and a∗ ∈ [Nj/Nl]. Therefore each b ∈(Aj ∩ Fl)⊕r is of the formb =l∑k=1b(k)Nk+b∗Nlwhere b(k) ∈ {r, . . . , rs} for k = 1, . . . , l and b∗ ∈ [rNj/Nl]. Hence|(Aj ∩ Fl)⊕r| ≤ (rs)lrNjNl= rl+1SlNjNl.(2.13.5)Now (2.13.2) follows by combining (2.13.3), (2.13.4), and (2.13.5).412.14 The Divergence of ‖f̂lµ‖p/‖fl‖L2(µ)In this section, we show thatliml→∞‖f̂lµ‖p‖fl‖L2(µ)=∞ ∀p ≤ p∗.(2.14.1)This is (2.0.4) in Theorem 2.0.3.Let l ∈ N0 and 1 ≤ p ≤ p∗ be given. Recall that r ∈ N was chosen so thatp∗ ≤ 2r. Since|f̂lµ(ξ)| ≤ µ(Fl) =SlTl∀ξ ∈ R,we have‖f̂lµ‖2r2r =∫R|f̂lµ(ξ)|2r−p|f̂lµ(ξ)|pdξ ≤(SlTl)2r−p‖f̂lµ‖pp.Therefore‖f̂lµ‖pp ≥(SlTl)p−2r‖f̂lµ‖2r2r.Since‖fl‖2L2(µ) = µ(Fl) =SlTl,it follows that‖f̂lµ‖pp‖fl‖pL2(µ)≥(SlTl)(p/2)−2r‖f̂lµ‖2r2r.42By combining (2.7.6) and (2.13.1), we have‖f̂lµ‖2r2r ≥ Cr(SlTl)2r Nlrl+1Sl.Therefore‖f̂lµ‖pp‖fl‖pL2(µ)≥ Cr(SlTl)p/2 Nlrl+1SlRecall that s = nα0−β0/2, t = nα0 , Sl = sl, Tl = tl, and Nl = nl. Note also thatrl = nlln rlnn . Therefore we have‖f̂lµ‖pp‖fl‖pL2(µ)≥ Crnl(− 14pβ0+1−α0+12β0−ln rlnn ).Recall now that n was defined so thatp∗ <2(2− 2α0 + β0)β0−4 ln rβ0 lnn.Using that p ≤ p∗ and rearranging, we get−14pβ0 + 1− α0 +12β0 −ln rlnn> 0.Thereforeliml→∞‖f̂lµ‖pp‖fl‖pL2(µ)=∞.This proves (2.14.1) and completes the proof of Theorem 2.0.3.43Chapter 3Explicit Salem Sets andApplications to MetricalDiophantine ApproximationIn this chapter, we generalize theorems of Kaufman [29] and Bluhm [5] to constructnew explicit examples of Salem sets in R. We also prove a multi-dimensional analogof our result and obtain new explicit sets in Rd with large Fourier dimension. Wegive applications of our results to metrical Diophantine approximation.In this chapter only, for x ∈ Rd,|x| = max1≤i≤d|xi|, |x|2 = (d∑i=1|xi|2)1/2.Let Q be an infinite subset of Z, let Ψ : Z → [0,∞), and let θ ∈ R. DefineE(Q,Ψ, θ) to be the set of all x ∈ R such that‖qx− θ‖ ≤ Ψ(q) for infinitely many q ∈ Q.Recall that, for x ∈ R, ‖x‖ = mink∈Z |x− k| is the distance from x to the nearest in-teger. We will always assume Ψ is bounded. Since ‖x‖ ≤ 1/2 for all x ∈ R, assuming44Ψ is bounded results in no loss of generality. We will also always assume Ψ(0) = 1.This assumption is imposed only to avoid tedious notation. Since redefining Ψ atfinitely many points does not change the set E(Q,Ψ, θ), assuming Ψ(0) = 1 resultsin no loss of generality.For M > 0, defineQ(M) = {q ∈ Q : M/2 < |q| ≤M} ,(M) = minq∈Q(M)Ψ(q).If Q(M) is empty, (M) is undefined.The main result of this chapter is the following theorem.Theorem 3.0.1. Suppose there is a number a ≥ 0, an increasing function h :(0,∞)→ (0,∞), and an unbounded set M⊆ (0,∞) such that|Q(M)|(M)ah(M) ≥Ma ∀M ∈M.(3.0.1)Then there is a Borel probability measure µ supported on E(Q,Ψ, θ) such that|µ̂(ξ)| . |ξ|−a exp(ln |ξ|ln ln |ξ|)h(4|ξ|) ∀ξ ∈ R, |ξ| > e.(3.0.2)Note the hypothesesM⊆ (0,∞) and (3.0.1) imply Q(M) is non-empty and (M)is defined for all M ∈M.We also have a higher-dimensional version of Theorem 3.0.1.Let m,n ∈ N, let Q be an infinite subset of Zn, let Ψ : Zn → [0,∞), and letθ ∈ Rm. Define E(m,n,Q,Ψ, θ) to be the set of all points(x11, . . . , x1n, . . . , xm1, . . . , xmn) ∈ Rmnsuch thatmax1≤i≤m‖n∑j=1qjxij − θi‖ ≤ Ψ(q) for infinitely many q ∈ Q.45Clearly E(1, 1, Q,Ψ, θ) = E(Q,Ψ, θ). As above, we will always assume Ψ is boundedand Ψ(0) = 1, and these assumptions result in no loss of generality.For M > 0, defineQ(M) = {q ∈ Q : M/2 < |qj| ≤M ∀1 ≤ j ≤ n} ,(M) = minq∈Q(M)Ψ(q).If Q(M) is empty, (M) is undefined.Theorem 3.0.2. Suppose there is a number a ≥ 0, an increasing function h :(0,∞)→ (0,∞), and an unbounded set M⊆ (0,∞) such that|Q(M)|(M)ah(M) ≥Ma ∀M ∈M.(3.0.3)Then there is a Borel probability measure µ supported on E(m,n,Q,Ψ, θ) such that|µ̂(ξ)| . |ξ|−a exp(ln |ξ|ln ln |ξ|)h(4|ξ|) ∀ξ ∈ Rmn, |ξ| > e.(3.0.4)Note the hypothesesM⊆ (0,∞) and (3.0.3) imply Q(M) is non-empty and (M)is defined for all M ∈M.We give some remarks on the proofs of Theorems 3.0.1 and 3.0.2 in Section 3.1.Sections 3.2 and 3.3 discuss motivations for Theorems 3.0.1 and 3.0.2. Section 3.4contains applications of Theorems 3.0.1 and 3.0.2 to metrical Diophantine approxi-mation. The combined proof of Theorems 3.0.1 and 3.0.2 constitutes Sections 3.5,3.6, and 3.7. Section 3.8 contains the proof of Lemma 3.3.1.3.1 Remarks on the Proof of Theorems 3.0.1 and3.0.2Since Theorem 3.0.2 is a generalization of Theorem 3.0.1, we give a single unifiedproof. The proof follows the methods of Kaufman [29] and Bluhm [5, 7]. We also46found Wolff’s variant of Kaufman’s proof in [49] instructive. Kaufman [29] andBluhm [7] considered E(Z,Ψτ , 0), where Ψτ (q) = |q|−τ and τ > 1. In [5], Bluhmconsidered E(Z,Ψ, 0) with Ψ(q) = ψ(|q|) and ψ : N→ (0,∞) decreasing. We neededrelatively straightforward modifications in our proof to handle the mn-dimensionalsetting, more general functions Ψ, and the case θ 6= 0. A more substantial modifica-tion was needed to handle general sets Q 6= Z. We now explain this modification.It is fairly easy to see that the proofs of Bluhm, Kaufman, and Wolff can beadapted to work for any set Q containing the set P of prime numbers. Their proofsuse two properties of the primes. The first property is about their density:|P ∩ (M/2,M ]| &MlnM∀M ∈ N.(3.1.1)The second property is about their arithmetic structure:|{p ∈ P : p divides n, p ≥M}| ≤lnnlnM∀n,M ∈ N, 2 ≤M ≤ n.(3.1.2)In our proof, we use the divisor bound of Wigert [48] (see Section 3.5) as a substitutefor (3.1.2). This allows us to handle more general sets Q that may not have anyspecial arithmetic structure. The cost is that we have the factor exp(ln |ξ|/ ln ln |ξ|)rather than ln |ξ| in Theorems 3.0.1 and 3.0.2. The analog in our proof of the densityproperty (3.1.1) is essentially the assumption (3.0.1) (though (3.0.1) is a combinedassumption about the density of Q and the size of Ψ).3.2 Motivation: Explicit Salem SetsThe first motivation for our main result is the construction of explicit Salem sets andexplicit sets with non-zero Fourier dimension. We start by recalling some definitionsand basic facts mentioned in Chapter 1.Let A ⊆ Rd. For α ≥ 0, the α-dimensional Hausdorff content of A isHα(A) = infB∑B∈B(diam(B))α,47where the infimum is over all countable collections B of balls such that A ⊆⋃B∈B B.The Hausdorff dimension of A, denoted dimH(A), is the supremum all of α ∈ [0, d]such that Hα(A) > 0. The Fourier dimension of A, denoted dimF (A), is the supre-mum of all β ∈ [0, d] such that|µ̂(ξ)| . (1 + |ξ|)−β/2 ∀ξ ∈ Rd.for some probability measure µ on Rd with supp(µ) ⊆ A. If A is a Borel set,Frostman’s lemma impliesdimF (A) ≤ dimH(A)(3.2.1)(cf. [34, Chapter 12], [49, Chapter 8]).A set A ⊆ Rd with dimF (A) = dimH(A) is called a Salem set. Every Borel set inRd of Hausdorff dimension 0 is a Salem set, every set in Rd that contains a ball is aSalem set of dimension d, and every sphere in Rd is a Salem set of dimension d− 1.Salem [41] proved the existence of Salem sets in R of arbitrary dimension α ∈(0, 1) using a random Cantor-type construction. Kahane [27] showed that for everyα ∈ (0, d) there is a Salem set in Rd of dimension α by considering the images ofcompact subsets of R under certain stochastic processes (cf. Chapters 17 and 18 of[28]). Moreover, Kahane [28, Chapter 18] showed that the image of any compactsubset of Rd under fractional Brownian motion is almost surely a Salem set, indi-cating that Salem sets are ubiquitous amongst random sets. Recently, other randomconstructions of Salem sets have been given by Bluhm [6] and Laba and Pramanik[31]. These random constructions do not produce explicit examples of Salem sets.Kaufman [29] was the first to find an explicit Salem set of dimension α /∈{0, d− 1, d}. The set Kaufman proved to be Salem is E(Z,Ψτ , 0), where Ψτ (q) =|q|−τ and τ > 1. If τ ≤ 1, then E(Z,Ψτ , 0) = R by Dirichlet’s approximation theo-rem, and so E(Z,Ψτ , 0) is a Salem set of dimension 1. We are concerned with τ > 1.48An easy and well-known argument (which we give in Section 3.8) givesdimH E(Z,Ψτ , 0) ≤21 + τ.Since E(Z,Ψτ , 0) is a Borel set, (3.2.1) impliesdimH E(Z,Ψτ , 0) ≥ dimF (Z,Ψτ , 0).Kaufman showed that for every τ > 1 there is a probability measure µ with supportcontained in E(Z,Ψτ , 0) ∩ [0, 1] such that|µ̂(ξ)| . |ξ|−1/(1+τ) ln |ξ| ∀ξ ∈ R, |ξ| ≥ 2.which impliesdimF E(Z,Ψτ , 0) ≥21 + τ,and hence that E(Z,Ψτ , 0) is a Salem set. See [7] for a detailed variation of Kaufman’sargument. In his thesis, Bluhm [5] showed that E(Z,Ψ, 0) is Salem for any Ψ withΨ(q) = ψ(|q|) and ψ : N→ (0,∞) decreasing. Technically, the results of Bluhm andKaufman are for E(N,Ψ, 0), not E(Z,Ψ, 0), but it is easy to adapt their proofs toE(Z,Ψ, 0).If A ⊆ R is a set of Fourier dimension α ∈ [0, 1], then it is easy to see the productset Ad ⊆ Rd has Fourier dimension at least α by considering product measures.A theorem of Gatesoupe [19] implies that if A ⊆ [0, 1] has Fourier dimension α ∈[0, 1], then{x ∈ Rd : |x|2 ∈ A}has Fourier dimension at least d− 1 + α. Moreover,Gatesoupe’s theorem implies that if A ⊆ [0, 1] is a Salem set of dimension α ∈ [0, 1],then{x ∈ Rd : |x|2 ∈ A}is a Salem set in Rd of dimension d − 1 + α. CombiningGatesoupe’s and Kaufman’s results yields explicit examples of Salem sets in Rd ofdimension α for every α ∈ (d− 1, d). Of course, the unit sphere in Rd and Rd itselfare explicit Salem sets with dimensions d− 1 and d, respectively. Explicit examplesof sets (Salem or otherwise) in Rd with Fourier dimension α ∈ (1, d−1) are unknown.Theorems 3.0.1 and 3.0.2 generalize the theorems of Kaufman [29] and Bluhm49[5]. Theorem 3.0.1 gives many new explicit Salem sets in R. Some particular Salemsets produced by Theorem 3.0.1 are discussed in Section 3.4. Theorem 3.0.2 givesnew examples of explicit sets in Rd with Fourier dimension α > 1.3.3 Motivation: Metrical Diophantine Approxi-mationThe second motivation for our main result comes from metrical Diophantine approxi-mation, where there is considerable interest in the Hausdorff dimension of E(Q,Ψ, θ).For τ ∈ R, define Ψτ : Z → [0,∞) by Ψτ (q) = |q|−τ . The classical Jarn´ık-Besicovitch theorem [26], [3] is thatdimH E(Z,Ψτ , 0) = min{21 + τ, 1}.In the setting of restricted Diophantine approximation, where Q is not necessarilyequal to Z, Borosh and Fraenkel [8] showed thatdimH E(Q,Ψτ , 0) = min{1 + ν(Q)1 + τ, 1},whereν(Q) = inf{ν ≥ 0 :∑q∈Qq 6=0|q|−ν <∞}.Eggleston [15] previously obtained this result for certain sets Q with ν(Q) = 0 orν(Q) = 1.There are also several results for more general functions Ψ. For Ψ of the formΨ(q) = ψ(|q|) with ψ : N→ (0,∞) decreasing, Dodson [14] showed thatdimH E(Z,Ψ, 0) = min{21 + λ, 1},50whereλ = lim infM→∞− lnψ(M)lnM.Hinokuma and Shiga [24] considered the non-monotone function Ψ(HS)τ (q) = |sinq||q|−τand proved thatdimH E(Z,Ψ(HS)τ , 0) = min{21 + τ, 1}.Dickinson [13] considered restricted Diophantine approximation with a function Ψsatisfying Ψ(q) = ψ(|q|) with ψ : N→ (0,∞) andλ = lim infM→∞− lnψ(M)lnM= lim supM→∞− lnψ(M)lnM.Dickinson deduced from the result of Borosh and Fraenkel above thatdimH E(Q,Ψ, 0) = min{1 + ν(Q)1 + λ, 1}.Rynne [40] proved a very general result that implies all of those above. Suppose onlythat Ψ : Z→ [0,∞) is positive for all q ∈ Q. Letη(Q,Ψ) = inf{η ≥ 0 :∑q∈Qq 6=0|q|(Ψ(q)|q|)η<∞}.Rynne showed thatdimH E(Q,Ψ, 0) = min {η(Q,Ψ), 1} .The main result in the case of inhomogeneous Diophantine approximation (i.e,the case where θ is non-zero) is due to Levesley [32]. Levesley showed that if Ψ(q) =ψ(|q|) with ψ : N→ (0,∞) decreasing, and ifλ = lim infM→∞− lnψ(M)lnM,51thendimH E(Z,Ψ, θ) = min{21 + λ, 1}.By an adaptation of Dickinson’s argument from [13], the assumption that ψ is de-creasing can be replaced by the assumption thatλ = lim infM→∞− lnψ(M)lnM= lim supM→∞− lnψ(M)lnM.The main content of the formulas above is the lower bounds they give on the Haus-dorff dimension of E(Q,Ψ, θ). The ≤-half of all the formulas for dimH E(Q,Ψ, θ)above are implied by the following lemma whose proof is well-known and straight-forward. For completeness, we give the proof in Section 3.8.Lemma 3.3.1.dimH E(Q,Ψ, θ) ≤ min {η(Q,Ψ), 1} .Because of (3.2.1), the Fourier analytic method of Theorem 3.0.1 stands as analternative to the usual methods of proving lower bounds on the Hausdorff dimensionof E(Q,Ψ, θ). In fact, Theorem 3.0.1 implies or implies special cases of all the resultsfor dimH E(Q,Ψ, θ) above (details are given in Section 3.4). Moreover, Theorem 3.0.1allows us to calculate the Hausdorff dimension of E(Q,Ψ, θ) in cases that (as far aswe know) have not been treated previously in the literature, such as the case whereθ 6= 0 and Q 6= N,Z.One particular advantage of the Fourier analytic method of Theorem 3.0.1 is theease with which it handles the inhomogeneous case θ 6= 0. In the proof of Theorem3.0.1 it is trivial to accommodate θ 6= 0, while Levelsey’s proof of his result for θ 6= 0is a non-trivial extension of Dodson’s proof for θ = 0.There are analogs of the formulas above for dimH E(m,n,Q,Ψ, θ). For example,Rynne [40] proveddimH E(m,n,Q,Ψ, 0) = min {m(n− 1) + η(Q,Ψ),mn}52when Ψ : Zn → [0,∞) is positive for all q ∈ Q andη(Q,Ψ) = inf{η ≥ 0 :∑q∈Qq 6=0|q|m(Ψ(q)|q|)η<∞}.Theorem 3.0.2 (via (3.2.1)) provides a lower bound on dimH E(m,n,Q,Ψ, θ), butit does not reach the true value of dimH E(m,n,Q,Ψ, θ) for any known case withmn > 1.3.4 ApplicationsIn this section we will present several consequences of Theorem 3.0.1 that give newfamilies of explicit Salem sets and imply formulas for dimH E(Q,Ψ, θ) discussed inSection 3.3. We will also present a typical consequence of Theorem 3.0.2 that yieldsexplicit sets in Rmn (mn > 1) with Fourier dimension larger than 1.Theorem 3.4.1. Assume Ψ is of the form Ψ(q) = ψ(|q|) with ψ : N → (0,∞) adecreasing function. Assume there is an increasing function h : (0,∞) → (0,∞)such that|Q(M)| ≥M/h(M) ∀M ∈ N(3.4.1)andlimx→∞lnh(x)lnx= 0Then E(Q,Ψ, θ) is a Salem set of dimension min {2/(1 + λ), 1}, whereλ = lim infM→∞− ln(ψ(M))lnM.(3.4.2)Proof. Since Ψ is bounded, λ ≥ 0.Let λ′ < λ and δ > 0. By (3.4.2), ψ(M) < M−λ′for all large M ∈ N. It follows53that∑q∈Qq 6=0|q|(Ψ(q)|q|)(2+δ)/(1+λ′).∑q∈Qq 6=0|q|−(1+δ) <∞.Therefore, by applying Lemma 3.3.1 and then letting λ′ → λ and δ → 0, we havedimH E(Q,Ψ, θ) ≤ min{21 + λ, 1}.If λ =∞, this argument shows dimH E(Q,Ψ, θ) = dimF E(Q,Ψ, θ) = 0.Assume 0 ≤ λ < ∞. Let λ′ > λ. By (3.4.2), there is an infinite set M ⊆ Nsuch that ψ(M) > M−λ′for all M ∈ M. Since ψ is decreasing, it follows that(M) > M−λ′for all M ∈M. Combining this with (3.4.1), we see that (3.0.1) holdswith a = 1/(1 + λ′). Since λ′ > λ is arbitrary, Theorem 3.0.1 givesdimF E(Q,Ψ, θ) ≥ min{21 + λ, 1}.Theorem 3.4.1 implies the result of Dodson [14] for dimH E(Z,Ψ, 0) discussedin Section 3.3. Theorem 3.4.1 also implies the formula for dimH E(Z,Ψ, θ) due toLevesley [32] mentioned in Section 3.3.Theorem 3.4.2. Assume Ψ is of the form Ψ(q) = ψ(|q|) with ψ : N → (0,∞).Assumeλ = lim infM→∞− lnψ(M)lnM= lim supM→∞− lnψ(M)lnM(3.4.3)and∑q∈Qq 6=0|q|−1 =∞.(3.4.4)Then E(Q,Ψ, θ) is a Salem set of dimension min {2/(1 + λ), 1}.54Proof. Since Ψ is bounded, λ ≥ 0.The same argument as in the proof of Theorem 3.4.1 showsdimH E(Q,Ψ, θ) ≤ min{21 + λ, 1}.If λ =∞, the argument shows dimH E(Q,Ψ, θ) = dimF E(Q,Ψ, θ) = 0.Assume 0 ≤ λ <∞. Seeking a contradiction, suppose|Q(M)| < M/ ln2(M)for all large M ∈ N. Then∑q∈Qq 6=0|q|−1 =∞∑k=0∑q∈Q(2k)|q|−1 .∞∑k=02−k∑q∈Q(2k)1.∞∑k=01ln2(2k)=1ln2(2)∞∑k=01k2<∞,which contradicts (3.4.4). So there is an infinite set M⊆ N such that|Q(M)| ≥M/ ln2(M) ∀M ∈M.Let λ′ > λ be given. By (3.4.3), ψ(M) > M−λ′for all large M ∈ N. Therefore(M) > M−λ′for all large M ∈ N. After removing finitely elements of M, we have(M) > M−λ′∀M ∈M.Then (3.0.1) holds with a = 1/(1 + λ′) and h(x) = ln2(x). Since λ′ > λ is arbitrary,Theorem 3.0.1 givesdimF E(Q,Ψ, θ) ≥ min{21 + λ, 1}.55Theorem 3.4.2 implies the result of Dickinson [13] for dimH E(Q,Ψ, 0) discussedin Section 3.3 in the case ν(Q) = 1. Consequently, it also implies the results ofBorosh and Fraenkel [8] and Eggleston [15] in the case ν(Q) = 1. Theorem 3.4.2implies the variation of the result of Levesley [32] for dimH E(Z,Ψ, θ) mentioned inSection 3.3 that uses Dickinson’s argument from [13].As far as we know, Theorems 3.4.1 and 3.4.2 represent the first calculation ofdimH E(Q,Ψ, θ) in the case where θ 6= 0 and Q 6= N,Z.Theorem 3.4.3. Suppose Ψ(HS)τ : Z → (0,∞) is defined by Ψ(HS)τ (q) = | sin q||q|−τ .Then E(Z,Ψ(HS)τ , θ) is a Salem set of dimension of min {2/(1 + τ), 1}.Proof. For every > 0,∑q∈Zq 6=0|q|(Ψ(q)|q|)2/(1+τ)+=∑q∈Zq 6=0|q|−1−(1+τ)| sin q|2/(1+τ)+ <∞.So, by Lemma 3.3.1,dimH E(Z,Ψ(HS)τ , θ) ≤ min{21 + τ, 1}.Let Q = {q ∈ N : | sin q| ≥ 1/2}. Clearlyminq∈Q(M)Ψ(HS)τ (q) ≥12M−τ ∀M ∈ N.It is also easy to see that|Q(M)| ≥M4pifor all largeM (for instance, by notingQ contains the nearest integer(s) to (2k+1)pi/2for every k ∈ N). It follows that (3.0.1) holds with a = 1/(1 + τ) and h(x) = 4pi.Therefore Theorem 3.0.1 and the fact E(Q,Ψ(HS)τ , θ) ⊆ E(Z,Ψ(HS)τ , θ) impliesdimF E(Z,Ψ(HS)τ , θ) ≥ min{21 + τ, 1}.56Theorem 3.4.3 implies the formula for dimH E(Z,Ψ(HS)τ , 0) due to Hinokuma andShiga [24] mentioned in Section 3.3.Finally, we give a typical consequence of Theorem 3.0.2 that yields explicit setsin Rmn with Fourier dimension larger than 1. The Hausdorff dimension of the setsis also determined for comparison.Theorem 3.4.4. Assume Ψ : Zn → [0,∞) is of the form Ψ(q) = ψ(|q|) with ψ :N→ (0,∞) andλ = lim infM→∞− lnψ(M)lnM= lim supM→∞− lnψ(M)lnM.(3.4.5)Assume h : (0,∞)→ (0,∞) is an increasing function such thatlimx→∞lnh(x)lnx= 0.(3.4.6)Assume there is an unbounded set M⊆ (0,∞) such that|Q(M)|h(M) ≥Mn ∀M ∈M.(3.4.7)ThendimH E(m,n,Q,Ψ, 0) = min{m(n− 1) +m+ n1 + λ,mn}anddimF E(m,n,Q,Ψ, 0) ≥ min{2n1 + λ,mn}.Proof. Since Ψ is bounded, λ ≥ 0.Let λ′ < λ and δ > 0. By (3.4.5), ψ(M) < M−λ′for all large M ∈ N. It followsthat∑q∈Qq 6=0|q|m(Ψ(q)|q|)(m+n+δ)/(1+λ′).∑q∈Qq 6=0|q|−(n+δ) <∞.57Since λ′ < λ and δ > 0 are arbitrary, we have∑q∈Qq 6=0|q|m(Ψ(q)|q|)η<∞ ∀η >m+ n1 + λ.Let λ′ > λ and δ > 0. By (3.4.5), ψ(M) > M−λ′for all large M ∈ N. Choose asequence (Mk)k∈N of numbers in M such that Mk ≥ 2Mk−1. Then∑q∈Qq 6=0|q|m(Ψ(q)|q|)(m+n−δ)/(1+λ′)&∑q∈Qq 6=0|q|−(n−δ) &∑k∈N∑q∈Q(Mk)|q|−(n−δ)&∑k∈NM−(n−δ)k |Q(Mk)| &∑k∈NM δk (h(Mk))−1&∑k∈NM δ/2k =∞Since λ′ > λ and δ > 0 are arbitrary, we have∑q∈Qq 6=0|q|m(Ψ(q)|q|)η=∞ ∀η λ. By (3.4.5), ψ(M) > M−λ′for all large M ∈ N.Therefore (M) > M−λ′for all large M ∈ N. After removing finitely many elementsof M, we have (M) > M−λ′for all M ∈ M. Combining this with (3.4.7), we seethat (3.0.3) holds with a = n/(1 + λ′). As λ′ > λ is arbitrary, Theorem 3.0.2 impliesdimF E(m,n,Q,Ψ, 0) ≥ min{2n1 + λ,mn}.583.5 Proof of Theorems 3.0.1 and 3.0.2: The KeyFunctionWe start with two basic definitions.Suppose f : Rd → C. If f ∈ L1(Rd), the Fourier transform of f is defined byf̂(ξ) =∫Rf(x)e−2piix·ξdx ∀ξ ∈ Rd.If f ∈ L1([0, 1]d) (i.e.,∫[0,1]d |f(x)|dx < ∞) and f is periodic for the lattice Zd (i.e.,f(x) = f(x+ k) for all x ∈ Rd and k ∈ Zd), the Fourier transform of f is defined byf̂(ξ) =∫[0,1]df(x)e−2piix·ξdx ∀ξ ∈ Rd.There is no ambiguity with these definitions; if f ∈ L1(Rd) and f is periodic for thelattice Zd, then f̂ = 0 using either definition.Now we begin the construction.Let K be a positive integer with K > mn+a. Let φ : Rm → R be a non-negativeCK function with supp(φ) ⊆ [−1, 1]m and∫Rm φ = 1. Then there is a C1 > 0 suchthat|φ̂(ξ)| ≤ C1(1 + |ξ|)−K ∀ξ ∈ Rm.(3.5.1)For > 0 and x ∈ Rm, let φ(x) = −mφ(−1x), andΦ(x) =∑k∈Zmφ(x− k).Note Φ is CK , is periodic for the lattice Zm, and satisfiesΦ̂(k) = φ̂(k) = φ̂(k) ∀k ∈ Zm.ThereforeΦ(x) =∑k∈Zmφ̂(k)e2piikx ∀x ∈ Rm59with uniform convergence.For q ∈ Zn, θ ∈ Rm, and x = (x11, . . . , x1n, . . . , xm1, . . . , xmn) ∈ Rmn, define xqby identifying x with the m× n matrix whose ij-entry is xij, and defineΦq,θ(x) = Φ(xq − θ) =∑k∈Zmφ̂(k)e2piik·(xq−θ).Note Φq,θ is CK and is periodic for the lattice Zmn.Assume from now on that qj 6= 0 for all 1 ≤ j ≤ n.If mn = 1, then for ` ∈ Z we haveΦ̂q,θ(`) =∑k∈Z∫[0,1]φ̂(k)e2piik(xq−θ)e−2pii`xdx=∑k∈Ze−2piikθφ̂(k)∫[0,1]e2piix(kq−`)dx={e−2piiq−1`θφ̂(q−1`) if q−1` ∈ Z,0 otherwise.In general, for ` = (`11, . . . , `1n, . . . , `m1, . . . , `mn) ∈ Zmn we haveΦ̂q,θ(`) =∑k∈Zm∫[0,1]mnφ̂(k)e2piik·(xq−θ)e−2pii`·xdx(3.5.2)=∑k∈Zme−2piik·θφ̂(k)∫[0,1]mne2pii(k·(xq−θ)−`·x)dx=∑k∈Zme−2piik·θφ̂(k)m∏i=1n∏j=1∫[0,1]e2piixij(kiqj−`ij)dxij={e−2piiq−11 `1·θφ̂(q−11 `1) if q ∈ Q˜(`),0 otherwise,whereQ˜(`) ={q ∈ Zn : q−11 `1 = · · · = q−1n `n ∈ Zm}and `j = (`1j, . . . , `mj) for all 1 ≤ j ≤ n.60For M > 0, defineFM(x) =1|Q(M)|∑q∈Q(M)Φ(M)q,θ (x) ∀x ∈ Rmn.This is the key function in the proof.Note FM is CK and periodic for the lattice Zmn. Since supp(φ) ⊆ [−1, 1]m and(M) = minq∈Q(M)Ψ(q), we havesupp(FM) ⊆ {x ∈ Rmn : max1≤i≤m‖n∑j=1xijqj − θi‖ ≤ Ψ(q) for some q ∈ Q(M)}.If q ∈ Q(M) and 0 < |`| ≤M/2, then q /∈ Q˜(`). So by (3.5.2) we haveF̂M(`) = 0 for 0 < |`| ≤M/2.(3.5.3)As φ ≥ 0 and φ̂(0) = 1, (3.5.2) also implies|F̂M(`)| ≤ F̂M(0) = 1 ∀` ∈ Zmn.(3.5.4)Choose i0 ∈ {1, . . . ,m} and j0 ∈ {1, . . . , n} such that |`| = |`j0| = |`i0j0|. By(3.5.1) and (3.5.2), for all ` ∈ Zmn we have|F̂M(`)| ≤ C11|Q(M)|∑q∈Q(M)∩Q˜(`)|φ̂((M)q−11 `1)|≤ C11|Q(M)|∑q∈Q(M)∩Q˜(`)(1 + (M)|q1|−1|`1|)−K= C11|Q(M)|∑q∈Q(M)∩Q˜(`)(1 + (M)|qj0|−1|`j0 |)−K≤ C1|Q(M) ∩ Q˜(`)||Q(M)|(1 + (M)M−1|`|)−K .61To bound |Q(M) ∩ Q˜(`)|, first note that for ` 6= 0 we haveQ˜(`) ⊆{(q1, . . . , qn) ∈ Zn :`i0j0qj0∈ Z, qj =`i0jqj0`i0j0∀1 ≤ j ≤ n}.It follows that|Q(M) ∩ Q˜(`)| ≤ |Q˜(`)| ≤ 2τ(|`|),where τ(v) denotes the number of positive integer divisors of the integer v. Thedivisor bound of Wigert [48] (cf. [21, p. 262]) sayslim supv→∞ln τ(v)ln v/ ln ln v= ln 2.For ζ > 0, definefζ(x) = exp(ζ lnxln lnx)∀x > e.So for any ζ > ln 2 there is a vζ ∈ N such thatτ(v) ≤ fζ(v) ∀v ∈ N, v ≥ vζ .Therefore|F̂M(`)| ≤ 2C1fζ(|`|)|Q(M)|(1 + (M)M−1|`|)−K ∀` ∈ Zmn, |`| ≥ vζ .(3.5.5)3.6 Proof of Theorems 3.0.1 and 3.0.2: The MainLemmaDefineg(ξ) ={1 if |ξ| ≤ e|ξ|−af1(|ξ|)h(4|ξ|) if |ξ| > eLemma 3.6.1. For every δ > 0, M0 > 0, and χ ∈ CKc (Rmn), there is an M∗ =M∗(δ,M0, χ) ∈M such that M∗ ≥M0 and|χ̂FM∗(ξ)− χ̂(ξ)| ≤ δg(ξ) ∀ξ ∈ Rmn.62The proof will show M∗ can be taken to be any sufficiently large element of M.Proof. Since χ ∈ CKc (Rmn), there is a C2 > 0 such that|χ̂(ξ)| ≤ C2(1 + |ξ|)−K ∀ξ ∈ Rmn.(3.6.1)For every p > mn, we havesupξ∈Rmn∑`∈Zmn(1 + |ξ − `|)−p <∞.(3.6.2)Since FM is CK and periodic for the lattice Zmn, we haveFM(x) =∑`∈ZmnF̂M(`)e2pii`·x ∀x ∈ Rmnwith uniform convergence. Since χ ∈ L1(Rmn), multiplying by χ and taking theFourier transform yieldsχ̂FM(ξ) =∑`∈ZmnF̂M(`)∫Rmnχ(x)e2pii(`−ξ)·xdx =∑`∈ZmnF̂M(`)χ̂(ξ − `)for all ξ ∈ Rmn. Then by (3.5.4) and (3.5.3) we haveχ̂FM(ξ)− χ̂(ξ) =∑|`|≥M/2χ̂(ξ − `)F̂M(`).(3.6.3)Case 1: |ξ| ≤ M/4. If |`| ≥ M/2, then |ξ − `| ≥ M/4 ≥ |ξ|. Hence by (3.5.3),(3.6.1), (3.6.2) and (3.6.3) we have|χ̂FM(ξ)− χ̂(ξ)| ≤ C2∑|`|≥M/2(1 + |ξ − `|)−K≤ C2(1 + |ξ|)−a(1 +M/4)−(K−a−mn)/2∑|`|≥M/2(1 + |ξ − `|)−mn−(K−a−mn)/2≤ δg(ξ)63for M sufficiently large.Case 2: |ξ| ≥M/4. Using (3.6.3), writeχ̂FM(ξ)− χ̂(ξ) =∑|`|≥M/2|`|≤|ξ|/2χ̂(ξ − `)F̂M(`) +∑|`|≥M/2|`|>|ξ|/2χ̂(ξ − `)F̂M(`) = S1 + S2.If |`| ≤ |ξ|/2, then |ξ − `| ≥ |ξ|/2 ≥ M/8. Hence by (3.5.3), (3.6.1), and (3.6.2) wehave|S1| ≤ C2∑|`|≥M/2|`|≤|ξ|/2(1 + |ξ − `|)−K≤ C2(1 + |ξ|/2)−a(1 +M/8)−(K−a−mn)/2∑|`|≥M/2|`|≤|ξ|/2(1 + |ξ − `|)−mn−(K−a−mn)/2≤δ2g(ξ)for M sufficiently large.Fix ln 2 < ζ < 1. By (3.0.1), (3.5.5), (3.6.1), (3.6.2) and because K > a we have|S2| ≤ 2C1C2∑|`|≥M/2|`|>|ξ|/2fζ(|`|)|Q(M)|(1 + (M)M−1|`|)−K(1 + |ξ − `|)−K≤ 2C1C2∑|`|≥M/2|`|>|ξ|/2|`|−afζ(|`|)((M)M−1)−a|Q(M)|(1 + |ξ − `|)−K≤ 2C1C2(|ξ|/2)−afζ(|ξ|/2)h(M)∑|`|≥M/2|`|>|ξ|/2(1 + |ξ − `|)−K≤δ2g(ξ)for all sufficiently large M ∈M.643.7 Proof of Theorems 3.0.1 and 3.0.2: The Mea-sureLet χ0 ∈ CK(Rmn) with∫Rmn χ0(x)dx = 1, supp(χ0) = [−1, 1]mn, and χ0(x) > 0 forall |x| < 1. With the notation of Lemma 3.6.1, defineM1 = M∗(2−1, 1, χ0), Mk = M∗(2−k−1, 2Mk−1, χ0FM1 · · ·FMk−1) ∀k ≥ 2.For k ∈ N0, define the measure µk bydµk = χ0FM1 · · ·FMkdx.By Lemma 3.6.1,|µ̂k(ξ)− µ̂k−1(ξ)| ≤ 2−k−1g(ξ) ∀ξ ∈ Rmn, k ∈ N.(3.7.1)Since g(ξ) is bounded , (3.7.1) implies (µ̂k)k∈N0 is a Cauchy sequence in the supremumnorm. Therefore, since each µ̂k is a continuous function, limk→∞µ̂k is a continuousfunction. By (3.7.1) we have| limk→∞µ̂k(ξ)− µ̂0(ξ)| ≤∞∑k=1|µ̂k(ξ)− µ̂k−1(ξ)| ≤ g(ξ)∞∑k=12−k−1 =12g(ξ)(3.7.2)for all ξ ∈ Rmn. Since µ̂0(0) = g(0) = 1, it follows from (3.7.2) that1/2 ≤ | limk→∞µ̂k(0)| ≤ 3/2.Therefore, by Le´vy’s continuity theorem, (µk)k∈N0 converges weakly to a non-trivialfinite measure µ with µ̂ = limk→∞µ̂k andsupp(µ) =∞⋂k=1supp(µk) = supp(χ0) ∩∞⋂k=1supp(FMk).65Because Mk ≥ 2Mk−1 and because (as we noted in Section 3.5)supp(FM) ⊆ {x ∈ Rmn : max1≤i≤m‖n∑j=1xijqj − θi‖ ≤ Ψ(q) for some q ∈ Q(M)},we havesupp(µ) ⊆ E(m,n,Q,Ψ, θ).Since χ0 ∈ CKc and K > a, we have |µ̂0(ξ)| . (1 + |ξ|)−a for all ξ ∈ Rmn. Combiningthis with (3.7.2) gives|µ̂(ξ)| . g(ξ) ∀ξ ∈ Rmn.By multiplying µ by a constant, we can make µ a probability measure. This completesthe proof of Theorems 3.0.1 and 3.0.2.3.8 Proof of Lemma 3.3.1Lemma 3.8.1 (Restatement of Lemma 3.3.1).dimH E(Q,Ψ, θ) ≤ min {η(Q,Ψ), 1} ,whereη(Q,Ψ) = inf{η ≥ 0 :∑q∈Qq 6=0|q|(Ψ(q)|q|)η<∞}.Proof. Since dimH E(Q,Ψ, θ) ≤ dimH R ≤ 1, we only need to provedimH E(Q,Ψ, θ) ≤ η(Q,Ψ).66Note E(Q,Ψ, θ) is invariant under translation by integers. ThereforedimH E(Q,Ψ, θ) = dimH⋃k∈ZE(Q,Ψ, θ) ∩ ([0, 1] + k)= dimH⋃k∈Z(E(Q,Ψ, θ)− k) ∩ [0, 1]= dimH⋃k∈ZdimH E(Q,Ψ, θ) ∩ [0, 1]= dimH E(m,n,Q,Ψ, θ) ∩ [0, 1].So it suffices to provedimH E(Q,Ψ, θ) ∩ [0, 1] ≤ η(Q,Ψ).Therefore, according to the definition of Hausdorff dimension, it will suffice to showthat for all > 0 and all η > η(Q,Ψ) there is a countable collection I of intervalsthat covers E(Q,Ψ, θ) ∩ [0, 1] and satisfies∑I∈I(diam(I))η < .Let > 0 and η > η(Q,Ψ). Define C = |θ|+ supq∈ZΨ(q). Observe thatE(Q,Ψ, θ) ∩ [0, 1] =⋂N∈N⋃q∈Q|q|≥N⋃k∈Z|k|≤C+|q|{x ∈ [0, 1] : |xq − θ − k| ≤ Ψ(q)} .Let N ∈ N. ThenE(Q,Ψ, θ) ∩ [0, 1] ⊆⋃q∈Q|q|≥N⋃k∈Z|k|≤C+|q|{x ∈ [0, 1] : |xq − θ − k| ≤ Ψ(q)}⊆⋃q∈Q|q|≥N⋃k∈Z|k|≤C+|q|Iq,k,67whereIq,k = [(θ + k −Ψ(q))/|q|, (θ + k + Ψ(q))/|q|].We have∑q∈Q|q|≥N∑k∈Z|k|≤C+|q|(diam(Iq,k))η =∑q∈Q|q|≥N∑k∈Z|k|≤C+|q|2η(Ψ(q)|q|)η≤∑q∈Q|q|≥N(2(C + |q|) + 1)2η(Ψ(q)|q|)η.∑q∈Q|q|≥N|q|(Ψ(q)|q|)η.The last sum converges because η > η(Q,Ψ). So, by taking N sufficiently large, wecan make the sum less than .68Chapter 4ConclusionIn this chapter, we summarize the contributions of this thesis and discuss directionsfor future work.4.1 ContributionsIn this section, we summarize the contributions of this thesis.The first major contribution of this thesis concerns the L2 Fourier restrictiontheorem due to Mockenhaupt [37] and Mitsis [35] (with the endpoint due to Bak andSeeger [1]). We state the theorem again here for convenience.Theorem 4.1.1. Let µ be a compactly supported probability measure on Rd. Supposethere are α, β ∈ (0, d) such thatµ(B(x, r)) . rα ∀x ∈ Rd, r > 0,(A)|µ̂(ξ)| . (1 + |ξ|)−β/2 ∀ξ ∈ Rd.(B)Then for all p ≥ 2(2d− 2α + β)/β we have‖f̂µ‖p . ‖f‖L2(µ) ∀f ∈ L2(µ).(4.1.1)Restriction theorems are some of the main objects of study in harmonic analysis.69They represent an important way to understand the nature of the Fourier transformand the geometry of sets and measures. Restriction theorems are connected to manyother important problems in mathematical analysis, including the Kakeya conjecture,the Bochner-Riesz conjecture, the estimation of solutions to the wave, Schroedinger,and Helmholtz equations, and the local smoothing conjecture for partial differentialequations (cf. [45]).Once a restriction theorem such as Theorem 4.1.1 has been established, it isnatural to ask whether it can be improved. In this thesis, we considered whetherwe can enlarge the range of p in the theorem by replacing 2(2d− 2α + β)/β with asmaller number. We considered this question only for a particular class of measures.If we do not limit the class of measures considered, the well-known Knapp example(cf. [49, Chapter 7]) shows the range of p cannot be enlarged. Whereas the classicrestriction theorems established before Theorem 4.1.1 applied only to smooth curvedsurfaces (like spheres) in Rd, d ≥ 2, the novel aspect of Theorem 4.1.1 is that itapplies to measures supported on fractal sets and to measures on R. In this thesis,we have considered the optimality of the range of p in Theorem 4.1.1 for measureson R. In particular, we proved the following theorem.Theorem 4.1.2. Given any 0 < β ≤ α < 1 and given any p∗ < 2(2 − 2α + β)/βthere is a compactly supported probability measure on R that satisfies (A) and (B)but does not satisfy (4.1.1) for every p ≤ p∗.The theorem is important and interesting because there is no obvious analog ofthe Knapp example (with its small, almost flat, spherical cap) available to prove theoptimality of the range of p in Theorem 4.1.1 when considering measures on the realline.The second major contribution of this thesis is the construction of new explicitSalem sets in R and new explicit sets in Rd with large Fourier dimension.There are many random constructions of Salem sets (cf. [41], [27], [28], [6], [31]).Moreover, Kahane [28, Chapter 18] showed that the image of any compact set inRd under fractional Brownian motion is almost surely a Salem set, which illustratesthat Salem sets are ubiquitous as random sets. These random constructions do not70produce any explicit examples of Salem sets. Moreover, besides the classic examplesof Salem sets in Rd of dimensions 0, d− 1, and d, explicit examples of Salem sets arevery difficult to find. The first (and essentially only) construction of explicit Salemsets with dimensions other than 0, d − 1, or d is due to Kaufman [29], who showedthat{x ∈ R : ‖qx‖ ≤ |q|−τ for infinitely many q ∈ Z}is a Salem set of dimension 2/(1 + τ) when τ > 1. Bluhm [7] generalized this byshowing that replacing |q|−τ by ψ(|q|) for any decreasing function ψ : N → (0,∞)still gives a Salem set. In this thesis, we further generalized the results of Kaufmanand Bluhm. For Q ⊆ Z infinite, Ψ : Z → [0,∞), and θ ∈ R, we proved a lowerbound on the Fourier dimension ofE(Q,Ψ, θ) = {x ∈ R : ‖qx− θ‖ ≤ Ψ(q) for infinitely many q ∈ Q} .Moreover, we showed that this set is Salem for many choices of Q and Ψ (regardlessof the value of θ). Explicit examples of sets in Rd with Fourier dimension larger than1 are also hard to find. We gave in this thesis some new explicit examples of suchsets. Furthermore, we applied our results to compute the Hausdorff dimension ofE(Q,Ψ, θ) in new cases, which is of interest in metrical Diophantine approximation.4.2 Directions for Future WorkIn this section, we discuss directions for future work related to the content of thisthesis.We first note that Chen [11] has obtained the following improved version of The-orem 4.1.2 by modifying the argument of Laba and myself from [23].Theorem 4.2.1. Given any 0 < β ≤ α < 1, there is a compactly supported proba-bility measure on R that satisfies (A) and (B) but does not satisfy (4.1.1) for everyp ≤ 2(2d− 2α + β)/β.There are several interesting ways Theorems 4.1.2 and 4.2.1 could possibly be71extended. We briefly discuss two of them.First, one could try to obtain a version of Theorem 4.1.2 or 4.2.1 with β > α. Sincethe Hausdorff dimension of a Borel set in R is the supremum of all α ∈ [0, 1] for whichthe set supports some probability measure µ satisfying (A), it is tempting to thinkthat the supremum of the numbers α ∈ [0, 1] for which a given probability measureµ satisfies (A) is the Hausdorff dimension of the support of µ. This is wrong. TheHausdorff dimension of supp(µ) can be strictly larger than the supremum of α ∈ [0, 1]for which µ satisfies (A). Moreover, since the Fourier dimension of Borel set in R isthe supremum of β ∈ [0, 1] for which the set supports some probability measure µsatisfying (B), it is tempting to think that the supremum of the numbers β ∈ [0, 1]for which a given probability measure µ satisfies (B) is the Fourier dimension ofthe support of µ. This is also wrong. The Fourier dimension of supp(µ) can bestrictly larger than the supremum of β ∈ [0, d] for which µ satisfies (B). Finally, sincethe Fourier dimension of a Borel set is always less than or equal to the Hausdorffdimension of the set, it is tempting to think a probability measure µ on R thatsatisfies (B) for some β must satisfy (A) for some α ≥ β. This, again, is wrong.A probability measure µ may satisfy (B) for some β and fail to satisfy (A) for anyα ≥ β. In other words, there are probability measures µ such that the largest αand β in (0, d) for which (A) and (B) hold obey β > α. Neither Theorem 4.1.2 norTheorem 4.2.1 give the sharpness of the range of p in Theorem 4.1.1 for the class ofsuch measures. It is likely a substantial new idea is needed to obtain such a sharpnessresult as it appears the current method cannot be modified to handle β > α.Second, it would be interesting to obtain a version of Theorem 4.1.2 or 4.2.1 formeasures on Rd and 0 < β ≤ α < d. The current method of proof does not seem togeneralize easily to this situation. However, it seems likely that such a theorem canbe obtained for α and β in a restricted range.Concerning the results in this thesis on the construction of explicit Salem sets,we pose three questions that are interesting for future research.What is the Fourier dimension of E(Q,Ψτ , 0) when ν(Q) < 1? For example,consider Q as the set of squares (so that ν(Q) = 1/2) or the set of powers of 2 (so thatν(Q) = 0). Theorem 3.0.1 says nothing when ν(Q) < 1. One natural conjecture is72that E(Q,Ψτ , 0) is a Salem set, meaning its Fourier dimension is min{(1+ν(Q))/(1+τ), 1}.What is the Fourier dimension of E(m,n,Z,Ψτ , 0)? Theorem 3.0.2 implies theFourier dimension is at least min {2n/(1 + τ),mn}. It is natural to conjecture thatthe Fourier dimension is exactly min {2n/(1 + τ),mn}. The verification of thisconjecture would make E(m,n,Z,Ψτ , 0) the first explicit example of a set in Rd(d ≥ 2) with Fourier dimension strictly between 1 and d − 1. Another natural con-jecture is that E(m,n,Z,Ψτ , 0) is a Salem set, meaning its Fourier dimension ismin{m(n− 1) + (m+ n)/(1 + τ),mn}.What is the Fourier dimension of E(m,n,Q,Ψ, θ) when no additional restrictionsare placed on the parameters? This is the most general question and therefore themost challenging.73Bibliography[1] J.-G. Bak, A. Seeger, Extensions of the Stein-Tomas theorem, Math. Res. Lett.18 (2011), no. 4, 767–781.[2] A. Banner, Restriction of the Fourier transform to quadratic submanifolds,Princeton University Thesis, 2002.[3] A. S. Besicovitch, Sets of fractional dimension (IV); on rational approximationto real numbers, J. London Math. Soc., 9 (1934), 126-131.[4] P. Billingsley, Convergence of Probability Measures, 2ed, Wiley (1999).[5] C. Bluhm, Zur Konstruktion von Salem-Mengen, Ph. D. Dissertation, Erlangen,1996.[6] C. Bluhm, Random recursive construction of Salem sets, Ark. Mat. 34 (1996),51–63.[7] C. Bluhm, On a theorem of Kaufman: Cantor-type construction of linear fractalSalem sets, Ark. Mat. 36 (1998), 307–316.[8] I. Borosh, A. S. Fraenkel, A generalization of Jarn´ıks theorem on Diophantineapproximations, Indag. Math. 34 (1972), 193-201.[9] J. Bourgain, Remarks on Montgomery’s conjectures on Dirichlet sums, Geomet-ric aspects of functional analysis (1989-90), 153-165, Lecture Notes in Math.,1469, Springer, Berlin, 1991.74[10] J. Bourgain, On the distribution of Dirichlet sums, J. Anal. Math. 60 (1993),21-32.[11] X. Chen, Sets of Salem type and sharpness of the L2-Fourier restriction theorem,preprint, 2013. http://arxiv.org/pdf/1305.5584v2.pdf, visited 2015-04-26.[12] M. Christ, Restriction of the Fourier transform to submanifolds of low codimen-sion, Thesis, U. Chicago, 1982.[13] H. Dickinson, A note on the theorem of Jarn´ık-Besicovitch, Glasgow Math. J.39 (1997), no. 2, 233-236.[14] M. M. Dodson, Hausdorff dimension, lower order and Khintchine’s theorem inmetric Diophantine approximation, J. Reine Angew. Math. 432 (1992), 69–76.[15] H. G. Eggleston, Sets of fractional dimensions which occur in some problems ofnumber theory, Proc. London Math. Soc. 54 (1951), 42–93.[16] C. Fefferman, Inequalities for strongly singular convolution operators, ActaMath. 124 (1970), 9–36.[17] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nded., Wiley (1999).[18] O. Frostman, Potentiel de´quilibre et capacite´ des ensembles avec quelques appli-cations a` la the´orie des fonctions, Meddel. Lunds Univ. Math. Sem. 3 (1935)1-118.[19] M. Gatesoupe, Sur un the´ore`me de R. Salem, Bull. Sci. Math. (2) 91 (1967),125-127.[20] B. Green, Roth’s theorem in the primes, Annals of Mathematics, 161 (2005),1609-1636.[21] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, 4th ed.,Oxford: Clarendon Press, 1975.75[22] K. Hambrook, Explicit Salem Sets and applications to metrical Diophantineapproximation, preprint. http://www.math.ubc.ca/~hambrook/research/Hambrook-Explicit-v2.pdf, visited 2015-04-26.[23] K. Hambrook, I. Laba, On the sharpness of Mockenhaupt’s restriction theorem,Geometric and Functional Analysis 23 (2013), no. 4, 1262-1277.[24] T. Hinokuma, H. Shiga, A remark on theorem of Jarn´ık, Ryukyu Math. J. 5(1992), 1-6.[25] W. Hoeffding, Probability inequalities for sums of bounded random variables,Journal of the American Statistical Association 58 (1963), no. 301, 13–30.[26] V. Jarn´ık, Diophantischen Approximationen und Hausdorffsches Mass, Mat.Sbornik, 36 (1929), 371–382.[27] J.-P. Kahane, Images d’ensembles parfaits par des se´ries de Fourier gaussiennes,C. R. Acad. Sci. Paris Sr. A-B 263 (1966) A678–A681.[28] J.-P. Kahane, Some Random Series of Functions, 2nd ed., Cambridge Univ.Press, 1985.[29] R. Kaufman, On the theorem of Jarn´ık and Besicovitch, Acta Arith. 39 (1981),265–267.[30] T. Ko¨rner, Hausdorff and Fourier dimension, Studia Math. 158 (2011), 37–50.[31] I. Laba, M. Pramanik, Arithmetic progressions in sets of fractional dimension,Geom. Funct. Anal. 19 (2009), no. 2, 429–456.[32] J. Levesley, A general inhomogeneous Jarnk-Besicovitch theorem, J. NumberTheory 71 (1998), 65-80.[33] W. Littman, Lp-Lq-estimates for singular integral operators arising from hyper-bolic equations, Partial Differential Equations (Proc. Sympos. Pure Math., Vol.XXIII, Univ. California, Berkeley, Calif., 1971), pp. 479-481. Amer. Math. Soc.,Providence, R.I., 1973.76[34] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Stud-ies in Advanced Mathematics, vol. 44, Cambridge University Press, 1995.[35] T. Mitsis, A Stein-Tomas restriction theorem for general measures, Publ. Math.Debrecen 60 (2002), 89–99.[36] G. Mockenhaupt, Bounds in Lebesgue spaces of Oscillatory integrals, Habilita-tionsschrift, U. of Siegen, 1996.[37] G. Mockenhaupt, Salem sets and restriction properties of Fourier transforms,Geom. Funct. Anal. 10 (2000), 1579–1587.[38] G. Mockenhaupt and T. Tao, Restriction and Kakeya phenomena for finitefields, Duke Math. J. 121 (2004), no. 1, 35–74.[39] E. Prestini, Restriction theorems for the Fourier transform to some manifoldsin Rn, Harmonic Analysis in Euclidean Spaces (Proc. Sympos. Pure Math. 35,Amer. Math. Soc., 1979, vol I), 101-109.[40] B. P. Rynne, The Hausdorff dimension of sets arising from Diophantine ap-proximation with a general error function, J. Number Theory 71 (1998), no. 2,166–171.[41] R. Salem, On singular monotonic functions whose spectrum has a given Haus-dorff dimension, Ark. Mat. 1 (1951), 353–365.[42] E. M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A.73 (1976), no. 7, 2174–2175.[43] E. M. Stein, Harmonic Analysis, Princeton Univ. Press, 1993.[44] R. Strichartz, Convolutions with kernels having singularities on a sphere, Trans.Amer. Math. Soc. 148 (1970), 461–471.[45] T. Tao, Some Recent Progress on the Restriction Conjecture, Fourier Analy-sis and Convexity, 217-243, Appl. Numer. Harmon. Anal., Birkhuser Boston,Boston, MA, 2004.77[46] P. A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math.Soc. 81 (1975), 477–478.[47] P. A. Tomas, Restriction theorems for the Fourier transform, in Harmonic Anal-ysis in Euclidean Spaces (Proc. Sympos. Pure Math. 35, Amer. Math. Soc.,1979, vol I), 111–114.[48] S. Wigert, Sur l’ordre de grandeur du nombre des diviseurs d’un entier, Ark.Mat. 3 (1906/7), 1-9.[49] T. Wolff, Lectures on Harmonic Analysis, Amer. Math. Soc., Providence, R.I.(2003).78Appendix AHoeffding’s InequalityThe standard version of Hoeffding’s inequality for real-valued random variables isthe following.Lemma A.0.1. If X1, . . . , Xn are independent real-valued random variables withEXi = 0 and |Xi| ≤ 1 for i = 1, . . . , n, thenP(∣∣∣∣∣1nn∑i=1Xi∣∣∣∣∣≥ u)≤ 2 exp(−nu2/2)for all u ≥ 0.For the proof of Lemma A.0.1, see [25].The following version of Hoeffding’s inequality for complex-valued random vari-ables can be deduced from Lemma A.0.1.Lemma A.0.2. If X1, . . . , Xn are independent complex-valued random variables withEXi = 0 and |Xi| ≤ 1 for i = 1, . . . , n, thenP(∣∣∣∣∣1nn∑i=1Xi∣∣∣∣∣≥ u)≤ 4 exp(−nu2/4)for all u ≥ 0.79Proof. For z ∈ C, if |z| = (|Re(z)|2 + |Im(z)|2)1/2 ≥ u, then |Re(z)| ≥ u/√2 or|Im(z)| ≥ u/√2. ThereforeP(∣∣∣∣∣1nn∑i=1Xi∣∣∣∣∣≥ u)≤ P(∣∣∣∣∣1nn∑i=1Re(Xi)∣∣∣∣∣≥u√2)+ P(∣∣∣∣∣1nn∑i=1Im(Xi)∣∣∣∣∣≥u√2).Applying Lemma A.0.1 on the right-hand side completes the proof.80Appendix BA Fourier Decay LemmaLemma B.0.1. Let µ be a complex-valued measure on R with compact support.Suppose there are positive constants b and c1 such that|µ̂(k)| ≤ c1(ln(e+ |k|))1/2(1 + |k|)−b ∀k ∈ Z.(B.0.1)Then there is a positive constant c2 such that|µ̂(ξ)| ≤ c1c2(ln(e+ |ξ|))1/2(1 + |ξ|)−b ∀ξ ∈ R.(B.0.2)Proof. If the support of µ is contained in an interval [−M,M ], then the measure νdefined by ∫Rf(x)dν(x) =∫Rf( 14M x+12)dµ(x) ∀f ∈ L1(µ)has support contained in [14 ,34 ] and satisfies ν̂(ξ) = µ̂(14M ξ)e−piiξ for all ξ ∈ R. There-fore it suffices to prove the lemma under the assumption that the support of µ iscontained in [14 ,34 ].Let γ : R→ C be a C∞ function that is equal to 1 on supp(µ) and has supp(γ) ⊆[0, 1]. For a ∈ [−1, 1], defineγa(x) = e−2piiaxγ(x) ∀x ∈ R.81We claim that for every n ∈ N there is a constant Cn,γ > 0 depending only on n andγ such that|γ̂a(k)| ≤ Cn,γ(1 + |k|)−n ∀k ∈ Z, a ∈ [−1, 1].(B.0.3)We now prove this claim. Fix n ∈ N. Note that for any compactly supported C∞function f : R→ C, integration by parts yieldsf̂ (n)(k) = (2piik)nf̂(k) ∀k ∈ Z.Thereforêγ(n)a (k) = (2piik)nγ̂a(k) ∀k ∈ Z, a ∈ [−1, 1].(B.0.4)By direct computation, the n-th derivative of γa isγ(n)a (x) = e−2piiaxn∑j=0(nk)(−2piia)jγ(n−j)(x) ∀x ∈ R.Therefore there exists a constant C ′n,γ > 0 depending only on n and γ such that|γ(n)a (x)| ≤ C′n,γ ∀x ∈ R, a ∈ [−1, 1].Since supp(γ(n)a ) ⊆ [0, 1] for every a ∈ [−1, 1], it follows that|̂γ(n)a (k)| ≤∫[0,1]|γ(n)a (x)|dx ≤ C′n,γ ∀k ∈ Z, a ∈ [−1, 1].Then (B.0.4) yields|γ̂a(k)| ≤C ′n,γ(2pi)n|k|−n ∀k ∈ Z, k 6= 0, a ∈ [−1, 1].82This implies (B.0.3) because (1 + |k|)n ≤ 2|k|n for k ∈ Z, k 6= 0 and|γ̂a(0)| ≤ ‖γa‖1 = ‖γ‖1 <∞ ∀a ∈ [−1, 1].Now that the claim is proved, we can complete the proof of the lemma. Letm ∈ Z and a ∈ [−1, 1] be given. Since γa is C∞ with supp(γa) ⊆ [0, 1], we haveγa(x) =∑k∈Zγ̂a(k)e2piikx ∀x ∈ R,where the convergence is uniform for x ∈ R. Then, as γ is equal to 1 on supp(µ), wehaveµ̂(m+ a) =∫Re−2piiaxe−2piimxdµ(x) =∫Re−2piiaxγ(x)e−2piimxdµ(x)=∫Rγa(x)e−2piimxdµ(x) =∫R∑k∈Zγ̂a(k)e2piikxe−2piimxdµ(x)=∑k∈Zγ̂a(k)∫Re−2pii(m−k)xdµ(x) =∑k∈Zγ̂a(k)µ̂(m− k).Writeµ̂(m+ a) =∑|k|≤ 12 |m|γ̂a(k)µ̂(m− k) +∑|k|> 12 |m|γ̂a(k)µ̂(m− k).We bound each sum separately. Fix a positive integer n > 1 + b and define Cb =∑k∈Z(1 + |k|)−n+b. If |k| ≤ 12 |m|, then12 |m| ≤ |m − k| ≤32 |m|, and so (B.0.1) and(B.0.3) give∑|k|≤ 12 |m||γ̂a(k)||µ̂(m− k)| ≤ c1Cn,γ(ln(e+ 32 |m|))1/2(1 + 12 |m|)−b∑|k|≤ 12 |m|(1 + |k|)−n≤ c1CbCn,γ(ln(e+ 32 |m|))1/2(1 + 12 |m|)−b.83Define C ′b = supx≥0(ln(e+ x))1/2(1 + x)−b. Then by (B.0.1) and (B.0.3) we have∑|k|> 12 |m||γ̂a(k)||µ̂(m− k)| ≤ c1C′bCn,γ∑|k|> 12 |m|(1 + |k|)−n≤ c1C′bCn,γ(1 +12 |m|)−b∑|k|> 12 |m|(1 + |k|)−n+b≤ c1CbC′bCn,γ(1 +12 |m|)−b≤ c1CbC′bCn,γ(ln(e+32 |m|))1/2(1 + 12 |m|)−b.Combining the bounds on the two sums yields|µ̂(m+ a)| ≤ 2c1CbC′bCn,γ(ln(e+32 |m|))1/2(1 + 12 |m|)−b≤ c1CbC′bCn,γ2b+ 32 (ln(e+ |m+ a|))1/2(1 + |m+ a|)−bfor every m ∈ Z and a ∈ [−1, 1]. This proves (B.0.2) because each ξ ∈ R can bewritten as ξ = m+ a for some m ∈ Z and a ∈ [−1, 1].It is possible to prove a more general version of Lemma B.0.1 (cf. [28, p. 252]).84