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Thin sets and strict-two-associatedness Hare, Kathryn Elizabeth 1986

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Thin Sets and Strict-Two-Associatedness By Kathryn Elizabeth Hare BMath., The University of Waterloo, 1981 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R OF P f f l L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Mathematics We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA March 1986 © Kathryn Elizabeth Hare, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M a t h e m a t i c s The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date A p r i l 24, 1986 DE-6(3/81) Abstract Let G be a compact, abelian group and let E be a subset of its discrete, abelian, dual group G. E is said to be a A(p) set if for some r < p there is a constant c(r) so that I I / I I P < c(r)||/||r whenever the support of / , the Fourier transform of / , is a finite subset of E. The main result of this thesis, Theorem 3.5, is that if E is a A(p) set, p > 2, and E satisfies a necessary technical condition, then for each 5 c G o f positive measure there is a constant c(S, E) > 0 so that \\lsf\\l>c(S,E)\\f\\l whenever / G L2(G) and support / C E. When such an inequality holds for all / G L2(G) with support f C E, then E and S are said to be strictly-2-associated. Actually we obtain the conclusion of strict-2-associatedness for a possibly larger class of sets than A(p) sets, p > 2, so that our theorem improves upon previously known work even when G is the circle group and Ed. Z. Most of Chapter 3 is dedicated to proving this result and showing that it is almost best-possible. In the remainder of Chapter 3 we establish necessary and sufficient conditions for a ii conclusion stronger than, but similar to strict-2-associatedness. In Chapter 4 we prove that if E is any A(p) set, p > 0, (or any set with the same arithmetic structure as A(p) sets) and if E satisfies the same necessary technical condition as in Theorem 3.5, then E is strictly-2-associated with all open subsets oiG. The proofs of these theorems depend on the arithmetic structure of A(p) sets. This topic is discussed in detail in Chapter 2. It has long been known that A(p) sets in Z with p > 2, cannot contain arbitrarily long arithmetic progressions and have "uniformly large gaps". We prove that no A(p) set, p > 0, can contain arbitrarily large parallelepipeds, a generalization of arithmetic progressions. This is new for A(p) sets, p < 1, in groups other than the circle. We introduce a definition which extends the notion of "uniformly large gaps" to the general setting. Combinatorial arguments are used to prove that sets which do not contain arbitrarily large parallelepipeds have this property. Finally, parallelepipeds are used to show that A(p) sets are built up from finite sets in a controlled way. This last fact and the notion of "uniformly large gaps" are central to the proofs we present of Theorems 3.5 and 4.1. iii Contents Abstract ii Acknowledgement v 1. A(p) Sets - Introductory Results 1 1.1 Notation and Terminology . 1 1.2 Definition and Examples of A{p) Sets 2 1.3 Uniformizable .4(2) Sets . 6 2. Arithmetic Properties of A(p) Sets 13 2.1 Introduction 13 2.2 Survey of Known Results 14 2.3 Parallelepipeds 18 2.4 The Uniformly Large Gap Property . 33 2.5 Mn Sets 38 3. Strict-Two-Associatedness for Uniformizable .4(2) Sets . . . . . . . . 44 3.1 Preliminaries 44 3.2 The Main Theorem 47 3.3 Very Strong-Two-Associatedness 67 4. Strict-Two-Associatedness With Open Sets 76 Open Problems 85 Bibliography 86 iv The author would like to express her gratitude to Dr. John Fournier for supervising this thesis and to her husband for his encouragement and continual support. She would also like to thank the Natural Sciences and Engineering Research Council and the I. W. Killam Foundation for their generous financial assistance. v Chapter 1 A(p) Sets - Introductory Results In this chapter we will introduce the terminology and basic definitions used in the remainder of the paper. As well, we present background information on A(p) sets and uniformizable v!(2) sets, 1.1 Notation and Terminology Throughout this thesis G will denote a compact abelian group equipped with the normalized Haar measure m, and J" will denote its necessarily discrete abelian dual group. The most familiar example of such a group G is the circle group T. Haar measure on T is normalized Lebesgue measure, and the dual of T is Z, the set of integers. For the group operation, our convention will be to use additive notation for Z and multiplicative notation otherwise. A standard reference for harmonic analysis on such groups is Rudin [19]. Trig(G) will denote the space of complex-valued trigonometric polynomials on G. For 0 < p < oo, LP(G) will denote the space of equivalence classes of 1 complex-valued functions, measurable with respect to m, and satisfying I I / I I P = ( / G l/| prfmy / P <oo . As usual we identify a function with the equivalence class to which it belongs. L°°(G) will denote the Banach space (of equivalence classes) of complex-valued functions, measurable and essentially bounded with respect to the measure m. If E is a subset of r, then a function / will be called an E'-function provided its Fourier transform, / , vanishes off E. A subscript B o n a space of functions restricts that space to its JS'-functions. For example, TrigE(G) consists of all E-polynomials. 1.2 Definition and Examples of A(p) Sets Definition 1.1 Let 0 < p < oo. A subset E of r is said to be a A(p) set if there is some r G (0, p) and a constant c(p, r) so that ll/ll P<c(p,r)||/ |L ior all f eTrigE(G). This notion was introduced by Rudin in [21] for subsets E of Z. An application of Holder's inequality as can be found in [21, 1.4] shows that if such a constant exists for some r G (0,p), then for each s G (0,p) there will be a 2 similar constant c(p, s). Holder's inequality also shows that if E is a A(p) set, then E is a A(t) set for all 0 < t < p. One could similarly define A(co) sets, but the only A(oo) sets in any abelian group are finite sets. Suppose p > 1 and that E C J 1 is a vd(p) set. Choose {ha} an approximate unit for Ll(G) consisting of trigonometric polynomials satisfying ||&a||i < 1. If / e LlE(G), t h e n f*h<*€ TrigE(G) so that ||/*Mp<e(p,l)||/*Mi . c fo i J i i/ iw iMi <c(p,l)||/||i. By Alaoglu's theorem the net {/*ha} has a weak* cluster point ^ € LP(G) satisfying IMIP<C(P,I)||/||I. Since lim ha(X) — 1, g(X) = f(X) for all X E T, thus by the uniqueness theorem a g = f. Because the converse is obvious, this establishes Proposition 1.2 E C r Is a A(p) set, for 1 < p < oo, if and only if there is a constant c(p) so that II/IIP<«(P)II/III 3 for all f e LE(G). As in [13, 5.3] an application of the Open Mapping Theorem proves Proposition 1.3 E C T is a A(p) set, for 1 < p < co, if and only if there is some q < p so that whenever f G LqE(G), then f G LP(G), and in this case the same statement holds for all q < p. Proposition 1.4 For 2 < p < co the following are equivalent: (1) E is a A(p) set; (2) there is a constant c(p) so that \\f\\p < c(p)||/||2 for all f G LE(G); and (3) whenever f G L%{G), then f G V>{G). The equivalence of (1) and (2) follow from Proposition 1.2 and Holder's inequal-ity, while the equivalence of (1) and (3) is a consequence of Proposition 1.3. Definition 1.5 E C r is called a Sidon set if there is a constant c so that for every feTrigE(G) E l / M I < 41/Hoo • xer A subset {nfc}£L, of Z is called a lacunary or Hadamard set if rik+i/rik >q>l for all A;. Lacunary sets are examples of Sidon sets. There are examples of Sidon sets which are not finite unions of lacunary sets (cf. [21, 2.5]). 4 Every Sidon set is a A(q) set for all q < oo. Indeed we have Theorem 1.6 [21, 3.1] If E is a Sidon set with constant c as in Definition 1.5, then for all f eTrigE(G), (1) H/llp < <Vp||/||2 if 2 < p < oo; and (2) U/lla < 2c|jjf||x • Condition (1) characterizes Sidon sets in the sense that if (1) holds for all p > 2 and all ^-polynomials / , then indeed E is a Sidon set [18]. In [3] Bonami constructed, for every infinite group J 1 , examples of sets which she showed were A(q) for all q < co. In [6] it is shown that these examples are non-Sidon sets. Rudin in [21, 4.8] constructed examples of subsets of Z which were A(2s) but not A(2s + e) for any e > 0, where s is any integer greater than 1. In certain other discrete abelian groups examples have been constructed of sets which were A(2s) but no better [7, 11]. Again s must be an integer greater than 1. There are no known examples of A(p) sets for p < 4 which are not already 1^(4) sets. It is known that if E is a A(p) set with 1 < p < 2, then E is a A(p + e) set for some e > 0 [1], but many open questions remain. For example: (1) Is every A(l) set a A(2) set? a A{4) set? (2) Is the union of two A(p) sets, p < 2, a A(r) set for some r > 0? For p > 2 this is obvious. Indeed the union of two l^(p) sets is another A(p) set. In Chapter 2, 5 in the comments after Corollary 2.20, we give some evidence for an affirmative answer to this question. 1.3 Uniformizable A{2) Sets An alternate characterization of A(p) sets, dual to Proposition 1.2, is Proposition 1.7 Let 1 < p < co and 1/p + 1/q = 1. E is a A(p) set, if and only if there is a constant c(p) so that corresponding to any function g G Lq(G) is a bounded function h satisfying (1) h(X) = g(X) for all X G E; and (2) HfclU < c(p)\\g\\q . Remark Proofs of this proposition may be found for subsets of Z in [21, 5.1] and for the general setting in [13, 5.3]. We give a proof here to illustrate the use of duality. Proof Suppose E is A{p) and | | / | | p < c(p)||/||1 for all / G LlE(G). Fix g G Lq(G) and define the linear mapping S : TrigE(G) —• C by / Hx)gjx) dm(x). JG 6 /Then \S(f)\<\\f\\MU<c(p)\\f\\i\\g\\g so by the Hahn-Banach Theorem S may be extended to a linear functional on Ll(G), also called S, with ||5|| < c(p)||g||g. By the Riesz Representation Theorem there is a bounded function h with S(f) = f f(x)hjx) dm(x) J G and IWU = < c{V)\\g\\q . Thus (2) holds for h. If X G E, the map x i-> X(x) belongs to TrigE(G) hence S{X)= f X(x)gjx) dm{x) =~gjxj. JG But also S(X) = / X(x)fc(z)dm(z) = h[X) JG so = h(x) for all X G E establishing (1). For the converse, let / G TrigE{G) and choose g G Lq(G) of g-norm equal to 1 so that II/IIP = / /(x)^)dm(x). * G 7 Choose the corresponding bounded function h satisfying (1) and (2) for this choice of g. Using the equality above, Parseval's relation and (1) we obtain n / i i p = E twm xer = E f(x)K(X) xer = f f(z)hjx)dm(x). JG By Holder's inequality and (2) \\f\\v<\\f\\mu<*mf\\i' Thus E is A(p). //// Suppose p = 2. Then a choice of g in L2(G) with £ 2 -norm equal to 1 and satisfying ll/lb = / f(x)gjx)dm(x) JG is g(x) = /(x)/||/||2, which belongs to L2E(G). This observation together with the proof of the previous proposition yields Corollary 1.8 E is A(2) if and only if there exists a constant c > 0 so that whenever g £ LE(G), there is a bounded function h with (1) h(X) = g{X) for all X e E; and (2) IHloo < c||<7||2 . 8 Following Blei [2], we define uniformizable .4(2) sets: Definition 1.9 E C r is said to be a uniformizable A(2) set if for each e > 0 there is a constant c(E, e) so that whenever g e L\ (G) there is a bounded function h with (1) h{X) = g{X) for all X 6 E; (2) |H|«> <c(£ ,e) | |<7 | | 2 ;and 1/2 (3) \\MrXj2=(Zxer\E\h(X)f) <s\\g\\2. The least such constant c(E,e) will be called the uniformizable A(2) constant for E and e. Remark Observe that (1) and (3) imply that \\h — <7||2<e||flf||2' It is immediate from Corollary 1.8 that uniformizable .4(2) sets are A(2) sets. Whether or not all A(2) sets are indeed uniformizable A(2) sets is an open problem. Since it is easy to see that the union of a uniformizable .4(2) set and a .4(2) set is another .4(2) set, a positive solution to this problem would answer the union question for .4(2) sets. Some important properties of uniformizable .4(2) sets are outlined in the next theorem. 9 Theorem 1.10 [9] For E C f the following are equivalent: (1) E is a uniformizable A(2) set. (2) For all e > 0 there is a constant Ci(E,e) so that for each g 6 LE(G) there is a bounded function h satisfying (a) WhWn^diE^WgW^and (b) \\h - g\\2 < s\\g\\2 . (3) For all e > 0 there is a constant c2(E,e) so that for each g e LE(G) there is a continuous function h satisfying (1), (2), and (3) of Definition 1.9 with this constant c2(E, e). (4) For all e > 0 there is a 8 > 0 such that whenever S is a measurable subset of G with m(S) < 6, then ( / s l / l 2 ) < * \ \ f h for all f e L%{G). (The family {/ e L\(G) : | | / | | 2 = 1} is said to be uniformly integrable.) (5) There is a Young function $ with — ^ • co as x —• co, for which we have LE(G) C L*(G), the Orlicz space corresponding to $ . Proof For completeness we sketch the proofs as they will appear in [9]. (1 => 2) and (3 1) are clear. 10 (2 => 3) Here we use the fact that given h G L°°(G) and e > 0 we can find a polynomial p with ||p||oo < Haloo and \\p - h\\2 < e. From this fact and (2) it follows that for each g G LE(G) there is a polynomial pi satisfying \\pi\\oo<c(E,S/4)\\g\\2 and - g\\2 < ^ \\g\\2 . Set <7i = g and inductively define a sequence {gn}%Li C L2E(G) by { 9^i - iC^i on E 0 otherwise and a sequence of polynomials {p„} satisfying ||Pn-<7„||2<|||<7„||2 and \\pn\U < c{E,e/4)\\gn\\2 . The continuous function h satisfying (3) is X^^Li pn • (2 => 4) Let e > 0 and put 8 = ———— . Given / G L%(G) choose h as in (2). Writing / as h + (/ — h) and applying the triangle inequality to j | l ^ / | | 2 , and then Holder's inequality, we see that if m(S) < 6 then ||ls/||2 < 2e||/||2 . (4 => 2) Given e > 0 and / G L\(G), let S = {x G G : > H/lb/v^}, where 6 is chosen as in (4). By Chebyshev's inequality m(5) < 6, thus if we let h = lscf, h satisfies (2) with Ci(E,s) = l/\/6 . (4 5) This can essentially be found in [20, 3.1]. / / / / 11 Remark In the proof of (5 => 4) the Closed Graph Theorem is used to ensure that the inclusion LE(G) «-* L®(G) is continuous. Thus there is a constant M with / $ ( | / | /M) < 1 for all / in the unit ball of L%{G). Corollary 1.11 Every A(p) set, p > 2, is a uniformizable A(2) set. Proof Either verify (4) or use (5) with = xp. //// This fact was first observed by Pisier. An alternate proof may be found in [2]. 12 Chapter 2 Arithmetic Properties of A(p) Sets 2.1 Introduction A(p) sets satisfy a number of arithmetic conditions which describe their "thin-ness" . This chapter is devoted to a discussion of these conditions. We begin by reviewing the well known relationship between A(p) sets in Z, p > 2; and arithmetic progressions; and extend these results to uniformizable A(2) subsets of Z. It easily follows that such sets have uniformly large gaps. We briefly discuss a generalization of this relationship for A(p) sets, p > 2, in arbitrary discrete abelian groups. Next, the notion of parallelepiped is introduced. We prove that no A(p) set, p > 0, in any discrete abelian group, may contain parallelepipeds of arbitrarily large dimension. This fact was previously known for all A(p) sets in Z, but in the general setting only for A(l) sets, and it can be used to obtain more results concerning arithmetic progressions. 13 We define a new arithmetic property, a generalization of the notion of uniformly large gaps, and prove that any set which does not contain parallelepipeds of arbit-rarily large dimension has this property. This new property will be used in proving our main result in Chapter 3. Finally, we extend a notion of Miheev's to the general setting and show that sets not containing parallelepipeds of arbitrarily large dimension are built from sets which have "large gaps" between any two members. This idea will also be used in Chapter 3. 2.2 Survey of Known Results The first arithmetic conditions related to A(p) sets are discussed in [21]. In that paper can be found the proofs of the next two propositions. Proposition 2.1 [21, 4.1] If E C Z is A(l), then E does not contain arbitrarily-long arithmetic progressions. Proposition 2.2(a) [21, 3.5] If E C Z is A(p), p > 2, and c = c(p,2) as in Definition 1.1, then whenever a, b £ Z, b ^ 0, and N is a positive integer, |{a + 6, a + 26,..., a + JV6} n E\ < 4c 2 JV 2 / p . Here | • | denotes the cardinality of the set. 14 Of course, the second proposition implies the first whenever E is a A(p) set for some p > 2. If E is only assumed to be a uniformizable A(2) set in Z a result similar to Proposition 2.2(a) is true. Indeed we have Proposition 2.2(b) If E c Z is a uniformizable A(2) set with constant c(E, e), then \{a + bia + 2bi.,.,a + Nb}nE\< 8(c(_7, e)2 +e2N). Proof We appropriately modify [21, 3.5]. Let KN{t) = J2n=-N (* ~ ) e ' n t b e t h e ^ ' t h F e J e r kernel. It is well known that \\KN\U = 1 and \\KN\\l < N. Consider the arithmetic progression A = {a + 6, a + 26,..., a + Nb}. Set N+l m = N/2 or — - — depending on whether N is even or odd, and define Q(t) = _ eimbtKN(bt)eiat. The choice of m ensures that Q(n) > 1/2 for n e A. Suppose ,4nI7 = {ni,..., ns} and let / be the ^-polynomial f(t) = _;£=1 e'nfct-Then | | / | | a = >/I a n d s r  s/2 < £ 0(n*)/(nfc) = / <?(*)/(*) <*m(«) . fc=i Since 2? is uniformizable .4(2) with constant c(Et e) there is an /i £ 2*°°(r) 15 satisfying \\h - f\\2 < e\\f\\2 and \\hl\n < c(JE?,e)||/||2. Thus s/2 < J {f - h)Q + JhQ <||/-%ilQ||2 + PI|oo||Q||i < eVsN + c(E,e)y/s. Hence \AnE\ = s < 8(s2N + c(E, e)2). //// An interesting consequence of Proposition 2.2(a) was observed by Miheev in [15], namely that A(p) sets in Z with p > 2 have "uniformly large gaps*. We make this notion precise and prove it for uniformizable A{2) sets. Corollary 2.3 Let E be a uniformizable A(2) set in Z and let N be any positive integer. There is an integer M = M(E, N) such that any interval of length M in Z contains a subinterval of length N free of points of E, that is, E has uniformly large gaps. I Proof Given E and N set e2 = Let M = c(E, e)2\6N. By Proposition 2.2(b), \{a,a+l,...,a + M - l } n E \ < 8{c(E,e)2 + s2M) for any a £ Z. Hence the interval [a, a + M — 1] of length M, must contain a M subinterval of length at least ^ £y _^ 2 ^ = ^ free of points of E. j j// 16 In Corollary 2.23 we obtain the same conclusion for any A(p) set, p > 0, in Z. In Definition 2.25 we generalize this notion to arbitrary discrete abelian groups. Definition 2.4 For positive integers d and N, Xi,... ,Xd G r and 1 < r < oo, define as in [13, 6.2] d d Ar(N,Xu...,Xd) = {[]>; ' : £l«y| r < Nr} • 3 = 1 j=l These sets may be viewed as generalized arithmetic progressions. Indeed, if r = Z and b € Z then Ar(JV,6) = {-JV6,...,-6,0, b,...,Nb}-is an arithmetic progression of length 2N + 1. Any arithmetic progression in Z of odd length is a translate of such a set. The next theorem is a generalization of Proposition 2.2(a). For a proof see [13, 6.3-6.4]. Theorem 2.5 Let E C T be a A(p) set with p>2 and let c = c(p, 2). Then \Ar(N,Xu...,Xd) n E\ < e V ( l + y/^N)2dlp for all Xi,... ,Xd e r, n G Z+ and r = 1,2 . 17 Again a slight modification, such as that given in the proof of Proposition 2.2(b), yields a corresponding conclusion whenever E is a uniformizable A(2) set. In Corol-lary 2.19 we present the proof of a similar statement for A(p) sets, p > 0. 2.3 Parallelepipeds Definition 2.6 A subset P of r is called a parallelepiped of dimension N if P is the product of N two element sets and P has exactly 2 elements. Thus P = H i l l * . ' . V>t} with Xi, ipi € T and all 2N terms are distinct. If A = {a, a + 6,..., a + (2^ — 1)6} is an arithmetic progression of length 2^ in Z, then A is a parallelepiped of dimension N, since A = {a, a + 6} + {0,26} + . . . + {0,2N~1b}. (In Z a parallelepiped of dimension N is a 2N element set which is the sum of N two element sets.) Thus parallelepipeds are another generalization of arithmetic progressions. Proposition 2.7 P is a parallelepiped of dimension N if and only if N = 2N and P = (J P, , where \Pt | = 2, (1) and for each 18 Proof If P is as in (1) then P = Pi Il i ls so P is a parallelepiped of dimension N. For the converse, observe first that if P is a parallelepiped of dimension 1 then P clearly satisfies (1). Now proceed by induction. Assume any parallelepiped of dimension N satisfies (1). If P is a parallelepiped of dimension N + 1, say P = niLt 1^" «^'}> * ^ e n certainly P' = f j f {Xi,i>i} is a parallelepiped of dimen-sion N. Thus by the induction assumption P' = UyLi-fy with |P{| = 2, and •^ = (Ui=in)7y,y = 2,3. . . ,^. Notice that every element of P is either an element of P'XJV+I or of P ' ^ + i • If we let PX = PIXN+I, 1 n + 1 = XJrl+l1>N+l and Py = ( I j t J P ,b / , J = 2,..., N + 1, then we have UyLi *V = -P'^JV+I and P N + L = P % + 1 , hence P = U ^ Y P J • T H U S P satisfies (1). //// In [16] Miheev makes the following definition: Definition 2.8 If {m,}^ C Z with mi > 0 and > mj + . ..mfc_i for A; = 2,3..., N, and if r € Z, then the spectrum of the trigonometric polynomial R(x) = eirx(l + eim*x)(l + eim'x) •••(! + eim»x) , i.e., the set {r,r + m1} + {0,m2} + {0,m3} + ... + {0,mN} , 19 I is called a tracing segment of length 2 . In [15] this is called a reflexive segment of length 2N. It is clear that any reflexive segment of length 2N is a parallelepiped of di-mension N in Z. Although not all parallelepipeds in Z are reflexive segments, any parallelepiped of dimension (2^ — 1) contains a reflexive segment of length 2^. To see this suppose P = Px + P2 + • • + P2N-i i s a parallelepiped of dimension (2^ - 1). By translating and reordering if necessary we may assume P, = {0,c,} with c , + i > c,- > 0 for all t = 1,..., 2N - 1. Now set 2 N - l mi = ci, m2 = c2 + C3, . . . , and — c,-. i-2N-l Clearly > m x + . . . + rrik-i for k = 2,..., N and the spectrum of (1 + e*'mi*)(l + eim>x) •••(! + eimNX) is contained in P. The main result of this section is: Theorem 2.9 1/ E is a A(p) set, p > 0, in any discrete abelian group, then there is an integer N such that E does not contain any parallelepipeds of dimension greater than N. 20 Remarks Before beginning the proof we discuss previously known results of this type. For p > 1 a proof can be found in [10]. The proof there is given for E C Z, but it directly generalizes to all discrete abelian groups. To obtain the conclusion for il(l) sets this proof makes use of the fact that any A{\) set is actually a A(\ + s) set for some e > 0 [1]. Other proofs are given for A(p) sets in Z, in [15] (p = 2) and [16] (p > 0). The main idea in the proof of [16] is used in Lemma 2.13 below. Proof of Theorem 2.9 Since any A(p) set with p > 1 is a A(s) set for any s < 1 we may without loss of generality assume p < 1. We will show in fact that N depends only on c(p,p/2). Since a translate of a A(p) set is a A(p) set with the same constant c(p,p/2), it suffices to show that A(p) sets do not contain parallelepipeds of the form P = Ili=i {IJ ^ i } ) \P\ = 2 M , for M> N. The proof will result by establishing a number of lemmas. Let us say that {Xi,... C P is quasi-dissociate if N = 1 f o r a = o,±i,i = i,...,N, implies e,- = 0 for all i = 1,..., N. Lemma 2.10 Fix a positive integer NQ and let Ni = SN° + 1. Any subset of r of cardinality Ni contains a quasi-dissociate subset of cardinality No. 21 Proof This is essentially an application of the Pigeon Hole Principle. Consider the subset {X,}^ C T. Choose Vi € {Xi,X2} so that Vi 7^ 1. If Ai = {ifrl1 : si = 0, ± 1 } then | A i | < 3 so it is possible to choose rp2 G {X,}*=1 with i>2 £ Ai. Now proceed inductively. Assume Vi > - - -»V'r* have been chosen. Let An = {Vi1^3 • • • : e,- = 0, ± 1 , t = 1,... n}. Since | A n | < 3 n we may choose Vn+i £ {X,-}?.*1 with Vn+i ^ -^ n • We may choose C {X,}^ in this way since JVi = 3No + 1. Now suppose n£L0i $V = 1 with, e* = 0, ±1, » = 1,..., No. Let A: be the largest integer with ek 7^  0. We cannot have A: = 1 for then rpf1 = 1 and hence rpi = 1. If A; > 1 then without loss of generality, ek = 1, so ipk = 11!=? ^T*'- *k*s imphes £ ^fc-i contradicting its selection. Thus e,- = 0 for all i = 1,2,..., N0 and hence {ipi}?~i is a quasi-dissociate set. / / / / Let us say that the parallelepiped Pu = f j ^ l j {1, X,-} 1 S (1) of order 2 if X? = 1 for i = 1,..., N; (2) dissociate if HiLi Xf1' = 1 with e,- = 0, ±1, ±2, implies e,- = 0 for all: = 1,..., N; and 22 (3) quasi-dissociate if HiLi^i' = 1 w ^ n si = 0 , ± 1 implies e,- '= 0 for all i = l , . . . , iV \ With this notation an immediate corollary of the previous lemma is Corollary 2.11 If E contains P = fl^ii!)*,}, 3 parallelepiped of dimension Ni = 3N° + 1, then E contains a quasi-dissociate, N0-dimensional parallelepiped. Next we will prove Lemma 2.12 Let E be a A(p) set, 0 < p < 1, with constant c(p,p/2). There is an integer Ni depending on c(p,p/2) such that E does not contain any parallele-pipeds of order 2 with dimension greater than Ni. 2(I-I/P)N0 Proof Choose an integer NQ so that 2N°'P = 2{I-2/p)N0 > C0°>P/ 2) 3 1 1 ( 1 s e t iVi = 3^° + 1. By Corollary 2.11 if E contains a parallelepiped of order 2 with dimension Ni then E contains a quasi-dissociate parallelepiped of order 2 with di-mension No, say n£L°i{l>Xi'} • Being quasi-dissociate and of order 2 the set {Xi}N^1 is probabilistically independent. Hence a N0 vl/p , N 0 f. \l/p nii+*.-r) = (n/ii+*.-n =2(i-i^N°. i=l ' \=1 ' 23 Similarly No \ 2/p ( / f l l 1 + x « i p / 2 ) = 2<1~2/p)JVo Thus if f(x) = nJL°i(l + Xi{x)), then / e TrigE(G) and H/IIP = 2^M*« > c(p)P/2)2^M^ = c(p,p/2)||/||p / a contradicting the fact that E is a .4(p) set with constant c(p,p/2). / / / / Lemma 2.13 Let E be a A(p) set, 0 < p < 1, with constant c(p,p/2). There is an integ-er N depending on c(p,p/2) such that E does not contain any dissociate parallelepipeds of dimension N. r v2 Proof It is shown in [16] that for any fixed r 6 (0,1) with — — < Choose so that AN > c(p,p/2)BN, and suppose E 1 contains the dissociate parallelepiped I l i l iOi^i} • Let R be the least solution of r = 2R 1 + R2 Let / = IlJLxCl + RXi). Then / G TrigE{G), and noi+^, iT / 2 ) H i + ^ 2 ( / n ( i + K ^ ) ) p / 2 ) 1 / p - (i) 24 An application of MacLaurin's formula shows that for any a G (0,1) {l + x)a = l + ax *——L— + Rem(x) where \Rem(x)\ < ( ** ^ )3 provided z G [~r,r] and r £ (0,1). Now - r < r ( * ' ^ ^ ^ ' ^ ^ < r so applying MacLaurin's formula to (1) with a = p/2 we obtain II/IIP > (l + R2)N/2(f I_(l + |r(^ ±5) - MLl r^2(^±5)2 = ( i + ( / nl1 - ^ H 5 ^ - (rb)3+ 1=1 2 v 2 ! ( i - f ) r y 2 + ; ^ y / p i=i = (i + ^,»/»n(.-iilri^_(_i_)»): because of the dissociateness assumption. Similarly ll/ll ,/ , < (i + a")"" n(i - 1 ^ S ^ + ( i r r ) ' ) ' " • «=1 Thus | | / | | p > (1 + R2)N'2AN > (1 + R2)N!2c(P>P/2)BN = c(p,p/2)\\f\\p/2 contradicting the fact that E is a A(p) set with constant c(p,p/2). / / / / 25 Lemma 2.14 For each positive integer N0 there is an integer N2 = N2(N0) so that if P = Il£L\ is a parallelepiped of dimension N2 with the property that for each i = l,2,...,N2 the set {j ^  i : X2 = X2} is empty, then P contains a dissociate parallelepiped of dimension NQ. Proof This is another application of the Pigeon Hole Principle similar to Lemma 2.10. / / / / Lemma 2.15 For each positive integer N0 there is an integer N = N(N0) so that if E contains a parallelepiped of dimension N, then a translate of E contains either a dissociate parallelepiped or a parallelepiped of order 2, with dimension N0 . Proof Fix N0. Put N = 2N0N2 with N2 = N2(N0) as in Lemma 2.14. Assume that a translate of E contains the JV-dimension parallelepiped P = n£Li{ljX»}. We will say that Xi ~ Xj if X2 = X2 . Let 5S- be the equivalence class containing Xi. We consider two cases. Case 1: For some 16 (1 ,2 , . . . , N}, \Si\ > 2NQ. Without loss of generality i = 1 and {1,2,..., 2N0} C Su i.e., X2k = X2 for A; = 1,2,..., 2JV0 . Then XxXk 1 = <pk satisfies <p\ = 1 for k = 1,..., 2N0 . Certainly IIy^i{^iV:,2i-i> Xi<P2j} C P and hence is a parallelepiped of dimen-sion N0 contained in E. A further translate of E contains the iV0-dimensional parallelepiped ITy i^{l> <P"2jtP2j-i} °f order two. 2 6 Case 2: Otherwise \Si\ < 2N0 for all i = 1,2,..., N. In this case there must be at least JV~2 distinct equivalence classes, say Si,...,5JV, . Lemma 2.14 may be applied to Il;=?i *° obtain a dissociate parallelepiped of dimension iV"0 in the original translate of E. Illl Completion of the Proof of Theorem 2.9 Put together Lemmas 2.12, 2.13 and 2.15. / / / / In the remainder of this chapter we discuss properties of sets which do not contain parallelepipeds of arbitrarily large dimension and thus obtain new results for A(p) sets. The results of Sections 2.4 and 2.5 will be used in Chapter 3. Definition 2.16 P C F will be called a pseudo-parallelepiped of dimension N if i 3 = Il£Li{X.-> where Xif ^ <E T. Note that a parallelepiped of dimension N is a pseudo-parallelepiped of dimen-sion N with cardinality 2N. Proposition 2.17 There are constants c(n) and 0 < s(n) < 1 so that if E C T does not contain any parallelepipeds of dimension greater than n, then \E n Pd\ < c(n)2deW 27 whenever Pd is a pseudo-parallelepiped of dimension d. This proposition is proved in [16] for E C Z and Pd a parallelepiped of dimen-sion d. Proposition 2.17 can be established by making appropriate modifications to this proof. We carry out similar modifications later, in the proof of Theorem 2.31. We will use this proposition to prove a series of results. Corollary 2.21 may be found in [16], as can Corollaries 2.18 and 2.20 for subsets of Z. Corollary 2.18 If E does not contain any parallelepipeds of dimension n and if AN is any arithmetic progression of length N, then \EnAN\ <2eWc(n)NeW where c(n) and 0 < e(n) < 1 are as in Proposition 2.17. In particular, if E C T is a A(p) set for any p > 0 then there are constants c(E) and 0 < e(E) < 1 so that \EnAN\ < c{E)NeW . (Compare with Proposition 2.2.) Proof Choose M such that 2 M _ 1 < N < 2M. Observe that the arithmetic progression AN — {X,Xtp,... ,XipN~1} is contained in the pseudo-parallelepiped {X,Xtl>} • nii2{l, V ' 2 ' " 1 } of dimension M . 28 Hence \E n AN\ < c(n)2M eW < 2eWc(n)NeW . //// Recalling Definition 2.4, it is natural to define d A0O{N,Xu...1Xd) = {T[X^ : sup |n,-| < N\ . Corollary 2.19 If E does not contain parallelepipeds of dimension n then \EnAr{N,Xi,...,Xd)\ <2d£Wc(n){2N + l)d£W for 1 < r < oo, where c(n) and 0 < e(n) < 1 are as in Proposition 2.17. (Compare with Theorem 2.5.) Proof Observe that d Ar(N,Xu...,Xd) C AT, Xd) = [ ] A^N^i). i=l But Aoo(N,Xi) is an arithmetic progression of length (2N+ 1), so n ? = i ^ ( J V , * , ) is contained in a pseudo-parallelepiped of dimension at most Md, where M is chosen so that 2M~l < 2N + 1 < 2M. 29 Hence \EnAr{N,Xu...,Xd)\ < c{n)2dMeW < 2<*ff(n)C(n)(2JV + 1)^") . //// Corollary 2.20 If Ei and E2 do not contain parallelepipeds of arbitrarily large dimension then neither does E\ U E2. Proof Assume Ei and E2 do not contain parallelepipeds of dimension n and let F/v be any parallelepiped of dimension N. Then u E2) n P N \ < \EX n PN\ + \E2 n PN\ < 2c(n)2Ne^ which is less than \PN\ if N is sufficiently large. / / / / Although this corollary by no means settles the union question for A(p) sets it does provide some evidence to support the belief in an affirmative answer. Corollary 2.21 The set of primes in Z is not a A(p) set for any p > 0. Indeed, if E = {nk}kLi C Z does not contain parallelepipeds of arbitrarily large dimension, then < 00 30 Proof If E does not contain parallelepipeds of arbitrarily large dimension then there are constants c and 0 < e < 1 so that \E D [2J, 2 J + 1 ) | < c2je since [2J',2J+1) is an arithmetic progression of length 23. Hence I n J = ^ ^ M < 0 ° 23 3=0 Illl For J? a subset of the positive integers and n G Z let r2(2£, n) be, as in [21], the number of ordered pairs {mi, m2} with rr\\, m 2 G i? and mi + m 2 = n. . In [17] Neugebauer showed that if E was a A(p) set in Z + with p = 2g' > 2, then if 1/q+l/q' = 1, \ i = l ' For p — 4, q' = 2 this result may be found in [21, 4.5]. Corollary 2.22 If E C Z + does not contain parallelepipeds of arbitrarily large dimension, then there is some p = 2q' > 2 and a constant c so that E satisfies the inequality < cN2'p 31 for all positive integers N. Proof Observe that if {a,-, 6,} is the set of pairs in E with a,- + 6,- = n, then a,-,6t- E (0,n] and if {a,-, 6,} and {ay,6y} are distinct pairs as above, then a,- ^ ay. Thus r2(E,n) < 2|(0,fi]D^| < 2nec where constants c and 0 < e < 1 are chosen as in Proposition 2.17. Hence N K l / q , N \ l /g n=l ' \ » = 1 ' < Nllq2Nec. Now JV 2/P - J V 1 - 1 / ' so that if — — < q < oo then p = 2q' > — i - > 2, hence 1 - <? 1 + e ( £ r 2 ( £ , n ) « ) < 2CAT2/P . llll Corollary 2.23 If E C Z does not contain parallelepipeds of arbitrarily large dimension then (a) E has uniformly large gaps, and /u\ TP L v _ •* • r ' |[o,o + J\T]n_J| ^ (bj .E- has zero uniform density i.e., lim sup J — '- = 0. N—oo a £ Z iV (Compare with Corollary 2.3.) 32 Proof (a) Imitate the proof of Corollary 2.3 replacing Proposition 2.2 by Corollary 2.18. Alternatively, (a) follows from (b); see for example the proof of Corollary 2.33. (b) Apply Corollary 2.18 to the arithmetic progression [a, a -f- N] of length N+l. //// Previously the zero uniform density of A(p) sets was readily seen for p > 2 as a result of Proposition 2.2. It was only known that ^l(l) sets had zero density, and this as a result of a difficult theorem of Szemeredi [22] on sets not containing arbitrarily long arithmetic progressions. The proof based on parallelepipeds not only gives a stronger conclusion but seems much easier. These corollaries show that sets which do not contain parallelepipeds of arbit-rarily large dimension satisfy the previously known necessary arithmetic properties of A(2) sets. 2.4 The Uniformly Large Gap Property Corollaries 2.18, 2.19, 2.22 and 2.23 use the notion of parallelepipeds to prove extensions in the spirit of previously known results. Our next goal will be to gener-alize the notion of uniformly large gaps to the setting of arbitrary discrete abelian 33 groups and show that sets which do not contain parallelepipeds of arbitrarily large dimension have this property. Toward this end we make the following definitions: Definition 2.24 Let F be a finite subset of T and X, ip G JH. We say that X is F-equivalent to rp if for some positive integer m there is a sequence X = Xi,.. .,Xm = ip with Xi+iX^"1 G F for t = 1,2,..., m — 1. Such a sequence will be called an F-chain joining X to rp. If Xi G E for t = 1,2,..., m, then X i , . . . , Xm will be said to be an F-chain in E joining X to tp, and in this case X will be said to be (E, F)-equivalent to tp. When F is a symmetric subset of F containing the identity this relation is an equivalence relation. The terminology is suggested by that of [13, 8.9]. If E C Z and F = [-N,N] then n < m €. E are (E, F)-equivalent provided the interval [n, m] does not contain any subinterval of length greater than N free of points of E. E has uniformly large gaps provided for every integer N there is an integer s > 0 with the property that if n — m ^  [—siV, sJV] then n and m are not (i?U{ji, m}, [—TV, iV])-equivalent. This has the obvious generalization stated below. 34 Definition 2.25 E C F has the uniformly large gap property provided for each finite, symmetric set F C r containing the identity, there is an integer s > 0 such that if Xip-1 £ Fs, then X and V are not (E U {X, ip}, F)-equivalent. Our next theorem extends Corollary 2.23(a) to the general setting with this interpretation of uniformly large gaps. Theorem 2.26 Suppose E C F does not contain any parallelepipeds of dimen-sion n and F is a finite symmetric subset of F containing the identity. There is a constant s = s(n, F) so that whenever {X,},e/ C F satisfy XiXJ1 £ Fs for all i ^  j, then X{ and Xy belong to distinct (Eu {X,},-6/,F)-equivalence classes when i ^ Thus E has the uniformly large gap property whenever E does not contain parallelepipeds of arbitrarily large dimension. In particular, all A(p) sets, p > 0, have the uniformly large gap property. Before proceeding with the proof we establish two lemmas. The second is mo-tivated in part by [23]. Lemma 2.27 Let 1 ^  cp E 2\ Suppose P' = {X,}?^ U^,-}?^ consists of2N+1 distinct elements of r with X,-^" 1 = <p for i = 1,2,.. .,2N. If one of {X,}^ or {V*»}?=i is a parallelepiped of dimension N, then P' is a parallelepiped of dimension N+l. 35 Proof Without loss of generality assume {X,} 2 = 1 = Pi • P 2 • • PN = P with |P,| = 2 and |P| = 2". Then P<p~l = { i^}-=x so P' = P x • • • PN • {1, v>~1}. llll Lemma 2.28 Let F be any finite subset of P. For each positive integer n there is a constant k(n) = k(n, F) so that if r > k(n) and {X,-}^ is an F-chain joining Xi and Xr, with Xi ^ Xj if i ^  j, then {Xi}j=1 contains a parallelepiped of dimension n. Proof Since any two element set is a parallelepiped of dimension one we may set fc(l) = 2. Now proceed inductively assuming the result for n. We consider the P-chain of distinct terms {Xi}ri=1 with r > 2fc(n)|P|fc(")-1 = k(n + 1). Notice that each of the sets Bl = {Xi}^, B * = {Xi}2i=k(n) + V B N = + ! where N = k(n+l)/k(n), form an P-chain of k(n) distinct terms, so by the induction hypothesis each contains a parallelepiped of dimension n. Observe that any two subsets of P, say A = {a,}™ t and B — {/?,'}£li> are translates of one another, i.e., A = BX for some X G P, if cti+iotj1 = /?f+1y0t~1 for all » = 1,..., m — 1. If in addition A n B = 0 and A contains a parallelepiped of 36 dimension n, then by Lemma 2.27 A U B contains a parallelepiped of dimension n+ 1. Since the set {X,}^=1 is an F-chain, there are only | F | choices for each of the characters Xi+iXf1. Thus there can be at most |F | f c ( n ) - 1 different sets of the form {Xi+iXt 1}i=\jLl)k(n)+i' y = 1,. • • ,^ V" • (Here we count different orderings as different sets.) But N was chosen to be twice this number, so at least two of the sets Bi,..., Bu must be translates. Their union, and hence {Xi}i-i, must contain a parallelepiped of dimension n+1. This completes the induction step. Illl We now prove the theorem. Recall that by assumption E does not contain parallelepipeds of dimension n . Proof of Theorem 2.26 We will show that if s = k(n,F) and X.XT 1 Fs for all i ^ j, then X,- and Xy belong to distinct (Eu {X,-},-g/, .^-equivalence classes when i ^ j. Suppose not. Then there is an F-chain in Eti{X,},^/, ipi,..., rpm, joining some pair X,-, Xy. Without loss of generality we may assume ^ 2 , • • •, V'm-i € E, i.e., Xy is the first occurence in the chain after X,- from possibly outside of E. If two of the characters ipk and ipi were equal, then the sequence ^ i i • • - > V"fc> V'J+i) • • • J VVn> 37 upon renumbering, would still be an P-chain joining Xi and Xj, so we may assume ^21 • • • > V ' m - i are distinct. Because tpi+iipj'1 e F for t* = 1,..., m - 1, it follows that Xj <E Fm~iXi and since F m _ 1 C Fs if m - 1 < s, we must have m - 2 > s = fc(n, P). Thus the P-chain in E, {ipi}1^1, consists of at least k(n, F) distinct terms and hence by the lemma E must contain a parallelepiped of dimension n. This contradiction establishes the theorem. / / / / Arguments similar to those above, but not given here, can be used to show that if E contains no parallelepipeds of dimension n, then the cardinality of any \p\k(n,F) _ j P-equivalence class in E is at most 1*1-1 2.5 Af n Sets Definition 2.29 A subset E of P is said to tend to infinity if for each finite subset A of P, there is a finite subset P of E such that if X, £ E \ F and X ^ ip then X V - 1 £ A -A subset E of Z tends to infinity if for each positive integer N only finitely many points of E differ in absolute value by at most N. Thus any lacunary set, for example, tends to infinity. 38 Any Sidon set is known to be a finite union of sets which tend to infinity. (See [13, 9.1] for countable Sidon sets and [4] for the general case.) Sets which tend to infinity can readily be shown to have special analytic prop-erties. For example: (1) Deschamps-Gondim [5, 5.1]: If E is a symmetric Sidon set which tends to infinity, then for every compact set K in G with non-void interior there is a finite set F C E and a constant c > 0 such that E l / M I <c||max(/, 0)1*1100 xer for all real-valued E\F polynomials. (2) Bonami [3, chap. IV]: If E is a A(4) set which tends to infinity and S is a subset of a connected group G with the Haar measure of S positive, then there is a constant c(S) > 0 such that I l/ l l 2<^)||/l 5||^ for all .^-polynomials / . The hypothesis of tending to infinity was shown by Deschamps-Gondim to be unnecessary in example 1. The conclusion of example 1 is in fact true for all Sidon 39 sets. For a presentation of this see [13, chap. 9]. We will show in Chapter 3 that it is unneccessary in example 2 as well. One of the ideas we use to prove this is to show that sets which tend to infinity are a central feature in the structure of all A(p) sets. In a sense all A(p) sets are built out of sets which tend to infinity and sets with large gaps between members. These ideas were introduced by Miheev in [15] for subsets of Z. We extend them to all discrete abelian groups. We define sets of class Mn inductively in the following manner: Definition 2.30 M 0 is the class of subsets of J 1 which tend to infinity. Mn is the class of subsets E of r which have the following property: for each finite set A C r, E can be expressed as E\ U E2, where X, x/; € E2, X ^ ip, implies X$~l ^ A, and Ei is a finite union of sets in class M n _ ! . In [15] Miheev shows that each class Mn contains a 4(4) set in Z which is not a finite union of sets in class M n _ i . He shows that any subset of Z which does not contain parallelepipeds of arbitrarily large dimension belongs to class Mn for some n. He also establishes that any sequence of integers which belongs to some class Mn has zero uniform density. A consequence of this fact, as is shown in Corollary 2.33, is that all subsets of Z which belong to some class Mn have uniformly large gaps. 40 This gives another proof that all A(p) sets in Z, p > 0, have the uniformly large gap property. First however, we illustrate how [15, Thm. 3] may be adapted to the general setting. Theorem 2.31 Every subset E of F which does not contain parallelepipeds of dimension n belongs to class M n _ 2 . In particular all A(p) sets, p > 0, belong to class Mn for some n > 0. Proof If E does not tend to infinity then for some finite subset A of r and each finite subset F of E, there are distinct members of E \ F, say X and rp, with Xrp~l € A . Since A is a finite set infinitely many of these pairs satisfy Xrp~l = <p for some <p £ A, <p ^  1. By Lemma 2.27 E contains a parallelepiped of dimension two proving the theorem for n = 2. Now proceed inductively assuming the result for n (> 2). Suppose that E contains no parallelepipeds of dimension n + 1. Let A be any finite subset of r. If only finitely many pairs {X, rp} C E, X ^ rp, satisfy Xrp-1 e A, then E satisfies the definition of class M „ _ 2 , (even Mo), with regards to A; so assume otherwise. As before E contains infinitely many two element sets P,- = {Xi,ipi} with X,-^,"1 = <Pi ^ A, <pi 7^ 1. Choose a maximal collection of these sets, say {P,-},-ej, subject to Pi D Pj = 0 for t ^ j. Then whenever Xip~* = <pi, X, rp E E, 41 at least one of X or rp belongs to a set in the maximal collection. By Lemma 2.27 neither of the sets {X,},e/ and {V'iK'e/ may contain any parallelepipeds of dimension n, since E was assumed to contain no parallelepipeds of dimension n +1. Applying the induction hypothesis we conclude that both sets belong to class M n _ 2 . Remove {X,},gj and { s^}ie/ from E. Notice that if X and rp belong to this new set, then Xrp~l ^ <pi, thus after repeating this process at most |A| times and deleting from E the (at most) 2|A| sets of class Af n _ 2 so obtained, there will be only finitely many pairs {X, rp), X ^  rp, remaining in E, with Xrp~l (E A . Since the addition of finitely many terms does not affect the class we may write E as E\ U E2, where Ei consists of finitely many sets of class M „ _ 2 , and whenever X,rp € E2, X 7^ rp, Xrp~l ^ A. Since A was arbitrary this implies that E belongs to class Of course, the converse to Theorem 2.31 is not true. There are subsets of Z which tend to infinity and contain parallelepipeds of arbitrarily large dimension, even arbitrarily long arithmetic progressions; for example M, n—l • Illl {1,2,4,7,10,13,17,21,25,29,34,39,44,49,54,...} . Proposition 2.32 // E C Z belongs to class Mn for some n, then E has zero uniform density. 42 A proof by induction on n can be found in [15, Thm. 2]. Corollary 2.33 If E C Z belongs to class Mn for some n, then E has the uniformly large gap property. Proof Actually we will prove that if a subset of Z has zero uniform density then it has the uniformly large gap property. Suppose this is not the case. Then there is a positive integer N with the property that for every M there is an interval of length M which does not contain a subinterval of length N free of points of E. These intervals must contain at least M/N terms from E. Hence \E(l[a,a + Mil limsup sup i — 1 > 1/N M - * o o a £ Z M contradicting the fact that E has zero uniform density. • / / / / As there are sets in class Mn which contain parallelepipeds of arbitrarily large dimension this result is more general than Corollary 2.23. Indeed the proof shows that any set with zero uniform density, such as the primes in Z, has uniformly large gaps. 43 Chapter 3 Strict-Two-Associateciness for Uniformizable 4(2) Sets 3.1 Preliminaries Definition 3.1 [13, 9.3] E C T is said to be strictly-2-associated with a mea-surable subset S of G provided there is a constant c = c(S,E) > 0 so that for all £"-polynomials / , \\lsf\\l>c(S,E)\\m. (1) In this case, inequality (1) holds for all / £ L2E(G). To see this choose a net {Pa} of -^polynomials converging in L2(G) to / G LE(G). For example, we could take for Pa, f * ha, where {ha} is a bounded approximate unit for Ll(G) consisting of trigonometric polynomials. Certainly ll^lslll - 11/1*112 and ||P„ls|g > e||P_||2 - c||/||2, hence (1) holds for / . 44 Any c > 0 which satisfies (1) for all ^-polynomials / will be called a constant of strict-2-associatedness for E and S. Our main result may be stated for connected groups, such as T, as follows: Theorem 3.2 Let G he a connected group. If E is a uniformizable A(2) set in r then E is strictly-2-associated with all measurable subsets of G with positive measure. Theorem 3.2 is a special case of Theorem 3.5, which will be stated and proved later in this chapter. In [15] Miheev obtained the conclusion of Theorem 3.2 for A(p) sets in Z with p > 2. As all A(p) sets with p > 2 are uniformizable A(2) sets, our work is a theoretical improvement even for subsets of Z. In [3] Bonami showed that if G was a connected group and E was a .4(4) set which tended to infinity, then E was strictly-2-associated with all measurable subsets of G with positive measure. As there are many known examples of ii(4) sets which do not tend to infinity, our work improves upon this result as well. We follow the general outline used by Miheev but give completely different proofs of most of the intermediate steps, as his proofs do not seem to adapt to uniformizable A(2) sets and/or to the general setting. On Z our proofs are simpler 45 than Miheev's. For the general setting we will use the uniformly large gap property discussed in Chapter 2. As is proven in Proposition 3.4 below, when G is not connected one can construct polynomials which vanish on certain subsets of G having positive measure. Thus in the general setting a further hypothesis is necessary. This hypothesis is related to the next definition which is given in [13, 8.2]. Definition 3.3 E C r is said to be X0-subtransversal, for Xo a subgroup of f, if whenever X, rp G E, X ^  rp, we have Xrp-1 £ X0 . This says that each coset of X0 intersects E in at most one point. Proposition 3.4 A subset E of T is X0-subtransversal for all finite subgroups XQ of r if and only if the only E-polynomial which can vanish on an open, non-empty subset ofG is the identically zero function. Proof (<=) If Xo is a finite subgroup of r then its annihilator G0 is open and non-empty, hence has positive measure. Suppose X, rp G E and Xrp~l G X0. The £"-polynomial X — rp vanishes on G0 and thus must be identically zero. Hence X = rp and E is Xo-subtransversal. 46 (=>•) A proof is given in [13, 8.12]. We prove a stronger result in Proposition 3.9. //// Thus in order for E to be strictly-2-associated with all subsets of G it is necessary that E be Xo-subtransversal for all finite subgroups X0 of J 1 . This is also the sufficent additional hypothesis we need to make the conclusion of Theorem 3.2 true in the general setting. 3.2 The Main Theorem Theorem 3.5 Let E be a uniformizable A(2) set in r. If E is Xo-subtransversal for all finite subgroups X0 of F then E is strictly-2-associated with all measurable subsets of G with positive measure. Before beginning the proof we make a few observations. Remarks If G is a connected group then r is a torsion free group, and {1} is the only finite subgroup of r. Thus any subset E of r is Xo-subtransversal for all finite subgroups X0 of JT, and so Theorem 3.5 reduces to Theorem 3.2 when G is connected. In [12] Lopez proves that all A(A) sets which tend to infinity and are Xo-sub-transversal for all finite subgroups XQ of r are strictly-2-associated with all mea-47 surable subsets of G with positive measure. A presentation of this may be found in [13, chap. 9]. The proof of Theorem 3.5 will comprise most of the rest of this chapter. Its organization is roughly as follows: (1) Show that whenever subsets of a uniformizable A(2) set are all strictly-2-asso-ciated with S C G with a common constant of strict-2-associatedness, and these subsets have sufficiently large "gaps" between them, then their union is again strictly-2-associated with S. This will be established in Lemma 3.6. (2) Use (1) together with the uniformly large gap property established in Chapter 2, to show that the union of a uniformizable A{2) set strictly-2-associated with S and a uniformizable A(2) set with large "distances" between members is stric-tly-2-associated with S. This step will be accomplished in 3.7 - 3.11. (3) Use the definition of class Mn as the union of a set with large "distances" between members and finitely many sets in class M n _ i to present an induction argument completing the proof of the theorem. Early in this process (Corollary 3.10) we will be able to extend the results of Bonami and L6pez to uniformizable A{2) sets which tend to infinity. Unless specified otherwise, E will denote a uniformizable A(2) set in J 1 , and S 48 will be a subset of G of positive Haar measure. Lemma 3.6 For each c > 0, there is a finite symmetric set F = F(E, S, c) C r, containing the identity, so that if the subsets {2£,} , e j of E are strictly-2-associated with S with c a constant of strict-2-associatedness for each, and EiEj1nF = d) for i^j, then Uie/ Ei 1 S a^so strictly-2-associated with S. Proof Since E is a uniformizable A{2) set there is a Young function $ and constant K so that | | / | | | < K\\f\\% whenever / £ L%(G), with 4>(z) - <}>{x2), <f> a "strongly convex" function (Theorem 1.10(5)). Without loss of generality suppose </> is itself a Young function and hence has conjugate ip. Define a Young function /?(x) = ip(x2). Let e = and choose a trigonometric polynomial P satisfying \\P-\s\\l<e. Let F = F(E, S, c) = (suppP^suppP)*1. Clearly F is a finite symmetric subset of r containing the identity. Suppose {Ei}i£i C E are strictly-2-associated with S with constant of strict-2-associatedness c, and satisfy EiEj1 n F = 0 for all i ^  j. We estimate Hls/H2. for any Uie/ ^ i-P°ly n o mi al / • Write / as X^ ie/ A' w^h /*' a n i^-polynomial for each 49 i G I. Observe that if for some X € T, 71P{X) ± 0 and frP{X)?0, then (suppfi^suppfj)-1 n F ? 0 . In particular EiEj1 n F / 0; so t = j . Hence II^ I^II = Since the sets {2£,-},-e/ are disjoint and strictly-2-associated with S with constant of strict-2-associatedness c, we have 41/112 = E c l l / .- l l 2 !<E H I^IS-Hence 41/111 < 2 EOK^ - ^ / . - l l i + ll^ /.-lll) <2E2||l5-P|||||/,-Hi + 2||P/||l <2E 2 £ l l/.-|ll + 2 | | P / | | ^ with the last inequality following from the choice of P. Now the functions /,• are ^ -polynomials, so | | / t | | | < if||/,||2. Thus we obtain c\\f\\l<AeKY,\\fi\\l + nPf\\l < 4sK\\f\\l + 2||P/||2 . 50 Similarly I I J V H a <2||(15-P)/||2 + 2||15/||2 <4,JR'||/||2 + 2||15/||2. Thus c||/|g<12eJr||/|g+4||ls/|g. Our choice of e implies l i z z i e > | I I / I B for all Uie/ ^ i - P ° l y n o m i a l s / J which completes the proof. / / / / For Ei,Ej C Z let d(Ei, Ej) = min{|nt- - n3 \ • n, G Ei} n3- £ Ej}, the size of the smallest gap between Ei and Ej. Lemma 3.6 says that if the sets {Et} C E C Z are strictly-2-associated with S with a common constant of strict-2-associatedness c, then there is an integer N = N(E, S, c) so that if d{E{, Ej) > JV for i ^  j, then U,e/ i s strictly-2-asso-ciated with S. Hence if the gaps between the sets {£",},•£/ are sufficiently large, their union is also strictly-2-associated with S. For the second step indicated in the outline of the proof of Theorem 3.5 we first establish 51 Proposition 3.7 Suppose E is strietly-2-associated with S. There is a finite subgroup X of r, depending on E and S, so that whenever the set Eu{X}, X E T, is X-subtransversal, then EL) {X} is s trie tly-2- associated with S, with constant of strict-2-associatedness independent of X. Indeed, if c is a constant of strict-2-associatedness for E and S, then E U {X} and S have as a constant of strict-2-associatedness where V is an open subset of G whose choice depends on E, S and c, and e is a constant determined byV. First we state and prove a preliminary lemma. is not identically zero. There is a finite subgroup XQ of J 1 and a constant e(g) > 0 so that if X, rp £ f and Xrp~l £ X0, then JG Proof Since g is not identically zero there is a 6 > 0 so that the measure of the set A — {x G G : g > 6} is positive. Lemma 3.8 Let g be a real-valued non-negative integrable function on G which Let X, rp G r. Observe that the integral Re lA(Xrp 1) < m(A) with equality if 52 and only if Xi> 1 = 1 on A, and hence on the smallest open subgroup containing A . Let X0 be the annihilator of this subgroup. If G is connected the only open subgroup of G is G itself and thus XQ would be trivial. In general, Xo is a finite subgroup of r and Rely^X^ - 1 ) = m(A) if and only if Xtf>-1 <E Xo . An application of the Riemann-Lebesgue Lemma yields an e > 0 so that Ref^XV' - 1 ) < (1 - s)m(A) whenever Xrp'1 £ X0. Thus, if Xip-1 ^ X0 we have j g\X-^>8 j l A \ X - ^ = s(2m(A) - 2ReU(XV'~1)) > 2e6m(A). Setting s(g) — e8m(A), the lemma is established. / / / / Proof of Proposition 3.7 Suppose c is a constant of strict-2-associatedness for E and S. Since E is a uniformizable A{2) set we can find by Theorem 1.10(4) a 8 > 0 so that whenever m(A) < 8 and / € LE{G), \\lAf\\l<\\\f\\l-The function v ^  m(5) - ls * ls-i(v) = mfS \ (SV 1 ) ) 53 is continuous; so there is a neighbourhood V of the identity in G with m(S\(Sv-1)) <6 whenever v £ V. Hence if Sv = (Sv~x) fi S and / is an ^ -polynomial then I|i * ./|g>l iwi i2 - i|i5\5 ./|g->l\\f\\l whenever v £ V. Given any E U {X}-polynomial / and v £ V, we follow Bonami [3] and let fv(x) = f(xv)-X(v)f(x). Observe that /.W) = /WW«)-x(»)) so that /„ is an ^-polynomial. The choice of Sv ensures that so that by applying the basic inequality 2(|a| 2 + |6|2)>|a + 6|2 54 to the line above and then using (1) and (2) we obtain 1 1 1 5 / 1 , 2 " 4m\v) Jvis \M*WimWimW = ^ y)\ E l / W I 9 / Mm -*(»)|2dm{v). We assume that EL) {X} is .X-subtransversal where X is the finite subgroup XQ chosen in Lemma 3.8 for the integrable function l v Then for all ip £ E, Xtp~l ^ X0, hence by Lemma 3.8 1 1 1 5 / 1 , 3 ^ 4 ^ V T I l / W ! a 2 « ( l v ) - (3) The basic inequality used before also shows that Ills/Ill >\J |/(X)X(x)|2 dm(x) - j | E / ( W ( z ) | 2 dm(x) rl>?x > \\f(x)\2m(s) - E I/MI9 • 4>*x (4) Thus by considering the two cases: 0) E ^ x l / ( V 0 l 2 > *II/II2; or (n) E ^ x \m? < 511/112 in which case |/(X)| 2 > (1 - 6)||/||2; for and substituting into (3) or (4), respectively, we obtain the constant of strict-2-55 associatedness given in the proposition with e = e(\y). Remarks Miheev in [15, Thm. 6] proved a similar result for subsets of Z, without obtaining a specific constant of strict-2-associatedness. His proof relied on special properties of Z. A weaker version of Proposition 3.7, without the requirement that the constant of strict-2-associatedness be independent of X, was proved for A(p) sets, p > 2, by Bonami for connected groups and L6pez for the general case. A presentation of this may be found in [13, chap. 9]. The same proof may be applied to uniformizable A(2) sets by making use of the uniform integrability property, Theorem 1.10(4). It is possible to prove Proposition 3.7 without obtaining a specific value for the constant of strict-2-associatedness by soft methods based on Bonami's result. We do not give the details here. Lemma 3.8 is a special case of the next proposition. For subsets of Z this proposition was known by Zygmund [24], and for the general setting a proof can be found in [13, 8.14]. We present here a new proof which is constructive. The technique is similar to the proof of Proposition 3.7. //// Proposition 3.9 Let a be a positive integer and g be a real-valued, non-negative integrable function on G which is not identically zero. There is a finite 56 subgroup X of T and a constant Si(cr,g) > 0 so that whenever f is a polynomial with suppf X-subtransversal and \suppf\ < cr, then f g\f\2dm>el(aig)\\f\\l. Proof As with the proof of Lemma 3.8 we may assume without loss of generality that g = Is for some measurable set S with m(S) > 0. If a = 1 the result holds trivially with ei = m(S). We proceed by indue-tion, assuming the result for all polynomials / with \suppf\ < a — 1 and suppf X-subtransversal for the appropriate subgroup X . As is seen in the proof of Proposition 3.7, it is possible to choose a neighbourhood V of the identity in G with V V ' 2(<r-l) for all v G V. Let Sv = Sv~x n S. Take for X the finite subgroup generated by the union of the finite subgroup given by the induction assumption for Is and cr — 1, and the finite subgroup Xo determined in Lemma 3.8 for Iv. Choose any X G suppf and let fv(x) = f{xv) — X{v)f(x) for each v G V. As in the proof of Proposition 3.7 we have fs l / | 2 d m " 4 ^ ) fy is l / 0 ( l ) | 2 d m W d m W • 57 Again fv(ip) = f{rp)(rp(v) - X(v)) so suppfv C suppf. Since \suppfv\ < a - 1, from the induction assumption (and choice of X) we have f \fv(x)\2 dm(x) = f \fv(x)\2 dm(x) - f \fv{x)\2 dm{x) J Sv JS JS\Sv~l >€X(a- 1, l5)||/,|g - \\fv\Lm{S \ Sv-1). But Thus ^ |/„(x)| 2 rfm(«) > - 1,1*)||/.H2 - i f - -l)S)) Hence > " ( g - 1 1 | 1 , ) i i / . i i i _«,(»-Ms) £ | / W p / w „ j _ x ( „ ) p < ( m ( „ ) MH £ - x ^ r " v ' - ^ ' - • ^ (5) the last inequality from Lemma 3.8 (again using the choice of X). But also J |/| 2dm > ^ - | / ( X ) | 2 - E l / M I * • (6) Again consider the two cases: 0) E ^ x l / ( V 0 I 2 > 511/112; or 58 00 E^x l/WI2 < 511/112; for 6 ~ m ( 5 ) I 2mjy) + m { S ) + V ' Substituting into (5) or (6), respectively, we obtain the conclusion of the proposition with 6si{a- l,lg)g(l v) 4m{V) llll Is) A consequence of Lemma 3.6 and Proposition 3.7 is Corollary 3.10 1/ E, in addition to being a uniformizable A(2) set, is X0-subtransversal for all finite subgroups X0 of r and tends to infinity, then E is strictly-2-associated with all measurable subsets of G with positive measure. Proof Let S C G have positive measure. Let F = F(E,S,m(S)) be the finite set from Lemma 3.6. Since E is assumed to tend to infinity there is a finite set A so that if X, V> e E \ A, X ^  rp, then Xrp-1 £ F. Now apply Lemma 3.6, taking as the sets Ei the singleton sets whose union is E \ A . Since the singleton sets are strictly-2-associated with S with constant of strict-2-associatedness equal to m(5), the choice of F ensures that E \ A is also strictly-2-associated with S. 59 Applying Proposition 3.7 |Aj times we conclude that E is strictly-2-associated with S. HI/ The uniformly large gap property is used now to complete step 2 of the outline of the proof of Theorem 3.5. Lemma 3.11 Suppose E is Xo-subtransversa] for all finite subgroups Xo of r and suppose E' C E is strictly-2-associated with S. Then there is a finite set Fi depending on E, E1 and S, so that whenever E" = {X,},ej C E satisfies XiXJ1 fi Fi if i j, then E' U E" is also strictly-2-associated with S. Proof Let c > 0 be a constant of strict-2-associatedness for all of the sets E' U {X}, X £ E, and choose the finite symmetric set F = F(E, S, c) containing the identity as in Lemma 3.6. Being a uniformizable A{2) set E does not contain parallelepipeds of arbitrarily large dimension, and hence E has the uniformly large gap property (Theorem 2.26). Choose the constant s so that if XiXJ1 fi Fs for i ^ j, and E" = {X,},e/, then the (E, F)-equivalence class containing X,- does not contain any other Xy G E". Denote by Ei the elements of this class which belong to E'uE". Set E0 = E'uE"\\Ji£IEi. We take for Fx the finite set Fs. For i G J, Ei C E'U {X,} while E0 C E\ hence {Ei}ieIU{0} are strictly-2-associated with S with constant of strict-2-associatedness c. By construction of the 60 equivalence relation, E{E-1 n F = 0 for i,j G / U {0}, i ^ j so by Lemma 3.6 U,e/u{o} E i = E'Li E" is strictly-2-associated with 5". / / / / Remark For the case when E is a subset of Z, Lemma 3.11 says that if E' C E is strictly-2-associated with S and {n*-} C E satisfies \nk — ny| > M for all k ^  j and for some sufficiently large M, then E'u{nk} is also strictly-2-associated with 5. Thus if we adjoin to a uniformizable A{2) subset of Z which is strictly-2-associated with S a uniformizable A(2) set with the property that the distance between any two members is sufficiently large, then this new set is still strictly-2-associated with S. As Theorem 2.31 says that E is built up inductively from sets which tend to infinity by sets with large distances between members, Lemma 3.11 is the tool we need to complete the proof of Theorem 3.5. Proof of Theorem 3.5 Recall from Theorem 2.31 that E belongs to class M/. for some k; thus it suffices to show that any subset of E which belongs to Mfc is strictly-2-associated with each set S of positive measure, for all positive integers k. We proceed by induction on k. Fix S and let E' be any subset of E which is strictly-2-associated with S. Suppose E" C E belongs to class Af0, i.e., E" tends to infinity. Choose the finite set Fi = Fi(E,E',S) by Lemma 3.11 so that whenever {X,},e/ C E satisfies XiXJ1 fi Fx for i,j G I,i ^ then E' U {X,},G/ is strictly-2-associated with S. 61 Since E" tends to infinity there is a finite set F so that if X, ip £ E" \ F and X ^  ip then XV*"1 fi Fi- Thus E' U {E" \ F) is strictly-2-associated with S. Applying Proposition 3.7 \F\ times we conclude that E' U E" is strictly-2-associated with S. Now suppose we have established that whenever E' C E is strictly-2-associated with S and 2?j£ C E belongs to class Affc, then E' U is also strictly-2-associated with S. Let E' C E be any set strictly-2-associated with S and assume that C E belongs to class Mk+i. Again choose the finite set Fi = Fi(E,E',S) as in Lemma 3.11. Since E'£+l belongs to class M^+i, it is the union of two sets Ei and E2, where if X, ip G Ex, X ^  V> then X ^ - 1 fi Fi, and is a finite union of sets in class Affc. From Lemma 3.11 we may conclude that E' U Ex is strictly-2-associated with S. By the induction hypothesis (E1 U Ex) U E2 = U is as well. This completes the induction step and hence the proof of the theorem. / / / / Corollary 3.12 Let E be a uniformizable A(2) set which is X0-subtransversal for all finite subgroups X0 of r. Let g G L2(G), g ^ 0. Then there is a constant c(g, E) > 0 so that for all f G L%(G), \\9f\\l>c{g,E)\\f\\l. 6 2 Proof Choose e > 0 and a subset S of G with positive measure, such that \g\ > e on S. Then \\9f\\l>e*\\lsm>s2c(S,E)\\f\\l f o r a l l / € l | ( G ) . / / / / . Corollary 3.13 Let E be a uniformizable A(2) set which is Xo-subtransversal for all finite subgroups XQ of T. If for some f 6 LE(G), m{x : f(x) = 0} > 0, then / = 0. Corollary 3.14 Suppose E is a uniformizable A(2) set and that either E is X0-subtransversal for all finite subgroups X0 of T or E tends to infinity. If f = YlxeE a*X converges pointwise on someset of positive measure, then f G L2(G). Proof By Egoroff's theorem there is a set S of positive measure on which Y2xeE a xX converges uniformly. If E tends to infinity then the proof of Corollary 3.10 shows that there is a finite set A C E so that E\A and S are strictly-2-associated. If E is X0-subtransversal for all finite subgroups XQ of r, then E itself is strictly-2-associated with S so let A = 0. For any finite set F let Sp — Ylxe{E\&)r\F axX- Then \\1S(SF - SF.)\\l > c(E, S)\\SF - SF.\\l 63 whenever F' is a finite subset of F, since SF — S'F E LE^A(G). Since Y^XEE axX converges uniformly on 5", and A is a finite set, {lsSV} is a Cauchy net (indexed by F) in L2(G). Thus {SF} is Cauchy in L2(G) and hence 52XEE\A axX E L2(G). Since A is a finite set, / e L2(G). //// Remark If r is a torsion group then the assumption that E tends to infinity is necessary to obtain the conclusion of the previous corollary. To see this suppose that E does not tend to infinity. Then for some finite set A C r there is an infinite set {(Xi, i>i)} C E of distinct pairs with X.V', 7" 1 £ A \ {1} . Without loss of generality we may assume A is a finite subgroup. Let S = A 1 - , the annihilator of A. Then S is an open set, hence a set of positive measure, but E»(X» — ipi) — 0 on S, while E , ( * . - V ' . ) ^ 2 ( G ' ) . Whether it is necessary for E to be a uniformizable 4^(2) set for Theorem 3.5 or these corollaries to be true is unknown. In 3.18-3.20 we discuss very strongly-2-associatedness, a conclusion stronger than strict-2-associatedness, which does imply that is a uniformizable A(2) set. We can show however that if E is strictly-2-associated with all subsets of G of sufficiently large measure then E must be a A(2) set. Thus Theorem 3.5 is almost best-possible. Indeed, if all A(2) sets are unifor-mizable A(2) sets then the hypotheses of Theorem 3.5 would be both necessary and sufficient for strict-2-associatedness. 64 Proposition 3.15 Let A be a subset of G with positive measure and let 0 < e < m(A). If E is strictly-2-associated with all subsets of A with measure at least m(A) — e, then there is a constant c(A) such that ||/U||i>c(ii)||/||a for all f e TrigE(G). When A — G the conclusion is that E is a A(2) set. An alternate proof of this case may be found in [8]. Proof If no such constant c(A) exists then there is a sequence { /„} C TrigE(G) such that | | / B | | 2 = 1 and | | / „ l A | | i < l /2 n . Let Bn = {x e A : |/B(x)| > 1/c}. By Chebyshev's inequality m(Bn) < e | | /„ lx | | i < e/2n. Thus if we let B = A \ | J n Bn, then m(B) > m(A) - e > 0. By hypothesis E is strictly-2-associated with B, contradicting the fact that / | / n | 2 < | | / n l B | | o o | | / n l B||l —> 0 as n —> oo . //// By duality we obtain the following corollary for uniformizable A(2) sets which 65 are X0-sub-transversal for all finite subgroups XQ of r. Corollary 3.16 If any set E C T is strictly-2-associated with all sufficiently large subsets of a subset A of G, then for such a set A there is a constant Ci(A) so that corresponding to each <f> € ^(E) there is a bounded function h supported on A with \\h\\oo < ci{A)\\<f>\\2 and h(X) = <f>{X) for all X€E. Remark The novelty here is that h is zero off A. Proof For fixed (f> £ P(E) consider the map S : {flA : / e TrigE(G)} C Ll{A) - C given by xer Since E is strictly-2-associated with A, the map S is well defined. By Proposi-tion 3.15 l ^ / U J I ^ I I / l b l H b ^ ^ l l / U l l x l H h , so that S is bounded. Now extend S to Ll(A), preserving the norm, by the Hahn-Banach Theorem. The proof is completed in the same manner as Proposition 1.7 by applying the Riesz 66 Representation theorem. //// Before turning to the notion of very strong-2-associatedness we state a partial converse to Proposition 3.15. Another proof of it can be found in [8]. Proposition 3.17 If E C r is a A(2) set, then E is strictly-2-associated with all subsets of G of sufficiently large measure. Proof Choose c so that ||/[| 2 < c||/||i for all / e L%(G) and suppose S C G has measure greater than 1 — 1/c2. If / G L2E(G) then Il/Ib^cll/Hi^cll/Isll!+ 011/1^11! < C | | / 1 5 | | 1 + C | | / | | 2 | | 1 5 C | | 2 , the last inequality by Cauchy-Schwarz. Transposing we obtain | | / | | a ( l - cm(fir«)i/a) < cH/l^lU . Since m(Sc)ll2 < 1/c we obtain the result. / / / / 3.3 Very Strong-Two-Associatedness We discuss in this section a property similar to, although stronger than, strict-2-67 associatedness with all subsets of positive measure. This property is only possessed by uniformizable A(2) sets. Definition 3.18 Let E be a subset of T and S a subset of G. E is said to be very strongly-2-associated with S if for each A > 1 there is a finite set F C E so that if / G TrigE^F{G) then A - M l / l l ^ ^ y l l W H ^ A| | / | |» . In other words, the average of | / | 2 over S is nearly its average over all of G. Proposition 3.19 E is very strongly-2-associated with all sets S of positive measure provided for each such set S and each A > 1 there is a finite set F C E so that if f e TrigE\F(G) then *-l\\f\\i<^\M\\i. (i) Proof Let S be a subset of G and A > 1. Certainly we may assume m(S) < 1. Notice that if (1) holds for some A > 1 then it also holds for all larger A, thus we may assume Am(5) < 1. mi Sc) If we set 6 = ——• then 6 > 1 and there is a finite set F so that 1 - m(S)X 1 IIWII^-'II/II2. m{Sc) 68 for all / G TrigE^F(G). For all such / we have I|i5/H2 = 11/112- II WII2 <(l-6-lm(Sc))\\f\\l = m(5)A||/||» as required. / / / / • Remark Similar arguments may be used to show that E is very strongly-2-associated with all sets S of positive measure provided for each such set S and each A > 1 there is a finite set F C E so that if / G TrigE^F(G) then It was shown by Zygmund in [25] that the lacunary sets in Z are very strongly-2-associated with all subsets of T with positive measure. Bonami in [3] showed that every A(4) set in r which tended to infinity has this property. In [9] Fournier obtained the same conclusion for uniformizable .4(2) sets which tend to infinity. We will prove that these hypotheses are both necessary and sufficient. Theorem 3.20 E C r is a uniformizable A(2) set which tends to infinity if and only if E is very strongly-2-associated with all subsets of G of positive measure. 69 1 IIWII 2 < AII/112. m ( 5 ) Proof Our proof of the sufficiency of the hypotheses will be proved by tech-niques similar to those used in Lemma 3.6. Fix S and A > 1. With the notation as in the proof of Lemma 3.6 choose a polynomial P with IIP , || < r _ ( l - A - 1 ) m ( 5 ) | |P | |oo<2 and | | P 2 - 1 5 | | ^ < £ . Let A = (suppP)(suppP)~~. Since E is assumed to tend to infinity there is a finite set P so that if X, ip E E\F, X ^  ip, then X ^ - 1 fi A. Suppose / E TrigE^F(G), say / = £ x , e £ \ F a i X i -As in the proof of Lemma 3.6 we have WfP\\l = £ I I « . * < P | | 2 = £ H 2 M = I I / H 2 H P I I Now , 2 ls\ But WfP\\l< j I / I 2 l 5 + / l / H P 3 < 11/1*112 + I I / I | 2 | | / ( P 2 - I 5 ) | | 2 | | / ( P 2 - l 5 ) | | 2 < v ^ | | / | | * | | P 2 - ls\\p < y/2Ke\\f\\2-since L%(G) C Hence Il / i 5|l2> 11/^112-v^^||/||2 > 11/115(11^ 113 - v^ATe) 70 Standard arguments show that m(S) < 3e+ | |P| | | . Thus \\fls\\l>\\m{m(S)-£(3 + V2K)) > \\f\\>(S)X - l This inequality combined with Proposition 3.19 shows the sufficiency of the hypotheses. Now we will establish the necessity of the hypotheses. First, suppose E does not tend to infinity. Then there is an infinite family ft of pairwise disjoint two-element sets {X, ip}, with X, ip € E and Xip~l = 7 7^  1. Let S — {x £ G : \~i(x) — Ij < 1/2}. S is an open, non-empty set so m(S) ^ 0. For each finite set F there is a pair (X,V0 e A, with X, ip £ E\ F, \\X - ip\\l = 2 Hence S does not satisfy the definition of very strongly-2-associated for any To complete the proof we show that E must be a uniformizable .4(2) set. This requires the following lemma whose proof we defer to the end of the proof of this theorem. and <\\{xrl-i)is\\i 00 < i / 4 ° A < 8. 71 Lemma 3.21 Suppose E is not a uniformizable A(2) set. Then there is an e > 0 so that for each finite set F and each positive integer n, there is a subset A = A(n, F) of G and an E \ F-polynomial gn satisfying m(A)<l/n and f |«7n|2 > e\\gn\\l . JA Proof of Theorem 3.20 (continued) Suppose E is not a uniformizable A(2) set. Choose e as in the lemma. Apply the lemma first with F equal to the empty set to obtain Ai C G and fx £ TrigE(G) with m(A\) < s/4 and / ! / i l 2 > * l ! / i l l 2 -Apply the lemma again with F = suppf\ to obtain A2 C G with m(A2) < e/8, A A and / 2 £ TrigE(G) with suppf2 n suppf\ = 0 and / \M2>4M1-Repeat this process to obtain sets An C G with m(An) < e/2n+1 and fn £ TrigE(G) with n - l supp/nfl ( [J suPPfi) =0 for n = 2,3,... and / \fn\2>e\\fn\\l J A-' n 72 Let S = nr=i^- Then oo m{S) = 1 - m( ( J 4 n ) > 1 - e/2 > 0 . n = l 1 — e/2 i£ and 5 are very strongly-2-associated; so given any fixed A with 1 < A < — 1 — £ there is a finite set F C E with the property that whenever / £ T r t g ^ ^ G ) , ^ l / I ^ A - ' m ^ H / l g Our construction of the functions fn ensures the existence of an integer N so that whenever n> N, fn 6 T r t ^ ^ G ) . Hence for all n> N, ^l / . | a > A - 1 m(W»|B>(l-«)||/.|g However / | / n | 2 < / i/«r = / l / n | 2 - / | /n| < (1 - s)\\fn\\l. This contradiction shows that E must indeed be a uniformizable A(2) set. / / / / We turn now to proving Lemma 3.21. First we will prove a weaker version. Lemma 3.21' If E is not a uniformizable A(2) set there is an e' > 0 with the property that for each positive integer n there is a subset An of G with m( An) < \/n 73 and an E-polynomial fn with J l/»|2 > *'ll/n|l2 . Proof This is the failure of the uniform integrability property for non-unifor-mizable A(2) sets (Theorem 1.10(4)). We remark that the E-functions gn in L2(G) which satisfy /' \9n\2 > e'\\gn\\l for some set An, m(An) < 1/n, may be assumed to be ^-polynomials, since TrigE(G) is dense in LE(G). //// Proof of Lemma 3.21 If E is not a uniformizable .4(2) set Lemma 3.21' holds for some s' > 0. Set s = s'/4. Assume the lemma is false. Then there is a finite set F, and a positive integer n so that whenever A C G satisfies m(A) < 1/n and / 6 TrigE^F(G), then jA l/l 2 < 41/112 = ^ 11/112. (i) Choose a positive integer N with ]Mmin^4W)' Let A be any subset of G with measure less than 1/N and let / be any A ^-polynomial. Denote by J\ the E \ F-polynomial YlxeE\F f{X)X. 74 By(l) , / j / i l 2 < ^ l l / i l l 2 < Now | | / - / l||oo< | F|||/||oo< | F | | | / | | 2 ) hence / \f?<2J \h\2 + 2f JA JA JA < y l l / l B + 2||/-/i||J0m(il) <(j + 2\F\*m{A))\\f\\l <A\f\\l e' since m{A) < . Since A and / were arbitrary this contradicts the weaker version of the lemma. //// 75 Chapter 4 Strict-Two-Associatedness With Open Sets The structural arguments used in Chapter 3 to prove Theorem 3.5 can also be used to prove Theorem 4.1 Suppose E C F is Xo-subtransversal for all finite subgroups X0 of F. If E belongs to class Mn for some n and E has the uniformly large gap property, then E is strictly-2-associated with all open, non-empty subsets of G. Before presenting the proof we mention some applications of this theorem. Corollary 4.2 Suppose E is Xo-subtransversal for all finite subgroups XQ of r and (1) E tends to infinity; or (2) E does not contain parallelepipeds of arbitrarily large dimension (in partic-ular, if E is a A(p) set for some p > 0); or 76 (3) E is a subset of Z and belongs to class Mn for some n. Then E is strictly-2-associated with all open, non-empty subsets of G. Proof (1) If E tends to infinity, then E belongs to class M 0 and clearly has the uniformly large gap property. (2) From Theorems 2.26 and 2.31 we know that E satisfies the hypotheses of the theorem. (3) All subsets of Z which belong to class Mn have the uniformly large gap prop-erty by Corollary 2.33. / / / / The following four lemmas will be used to prove Theorem 4.1. The first is a standard topological argument. Lemma 4.2 Let U C G. If S is any open set which contains the closure of U then there is a neighbourhood V of the identity such that U - V C S. Proof For each u £ U, choose an open set Nu containing u such that Nu C S. Choose a neighbourhood of the identity Vu so that V t t • Vu C iV" uu _ 1 . The open sets {Vuu}uejj cover U hence we may choose a finite subcover {VuMi}^l.l. Let v = nr=i vUi. If u € U and v G V then u £ V^.u,- for some i = 1,2,...,m, and v £ VUi, thus uv £ VUiUiVUi C Nu.u,u;1 c s. mi 77 Lemma 4.3 Suppose E C r and U is an open, non-empty subset of G such that E and U are strictly-2-associated. Let S be any open subset of G containing U. There is a finite subgroup X0 of r, depending on S and E, so that whenever the set E U {X}, X € T, is XQ-subtransversal, then E U {X} is strictly-2-associated with S, with constant of strict-2-associatedness independent of X. (Compare with Proposition 3.7.) Proof Choose a neighbourhood V of the identity so that UV C S. Given v EV and / G L2Eur JG), let fv{x) = f(xv) - X{v)f(x). Observe that If we assume \\UJf\\% > C i | | / | | | whenever / (E Tr\gE(G), then since /„ G TrigE(G) for all v G V we have The proof of the lemma is completed in the same manner as was the proof of WUfWl = ^ y)lvls l / W I 2 dm(x)dm(v) Proposition 3.7. //// Corollary 4.4 Suppose E C F is X0-subtransversal for all finite subgroups X0 of r. If there is a finite set F and an open, non-empty set U so that E\ F 78 and U are strictly-2-associated, then E is strietly-2-associated with any open set S containing U. Proof Assume F = {Xi,.. -,XN}- Being a compact Hausdorff space, G is normal. Thus it is possible to choose open sets Si,,..,S^-i satisfying U~CSiCS~i~cS2C...C SV-i C S . By the previous lemma (E \ F) U {Xi} is strictly-2-associated with Si, hence (E \ F) U {XI,X2} is strictly-2-associated with S2, and so by induction E is stric-tly-2-associated with S. //// Lemma 4.5 Let S and Si be open, non-empty subsets of G with Si C S. Given c > 0 there is a finite, symmetric set F = F(c,Si,S) C F, containing the identity, so that if the sets {F,},e/ C r are strictly-2-associated with Si, with con-stant of strict-2-associatedness c, and E{Ejl n F = 0 for all i,j 6 7, t / then U t g / Ei is strictly-2-associated with S. (Compare with Lemma 3.6.) Proof Since G is normal there is a continuous function g : G —• [0,1] with g(Si) = 1 and g(Sc) — 0. Let P be a polynomial with HP - gWl < s = c/12 . (1) 79 Set F = (suppP)(suppP)-1. As in Lemma 3.6, if / <E TriguEi(G), f = £,-e/ /, with /,• e TrigE.(G), then \\pf\\l-Y^\\pfi\\ Thus \\isf\\l>\\gf\\l >\\\Pf\\l - \\{P~9)f\\l > \ E wp^\l - *\\f\\l the last step by an application of Holder's inequality and (1). But also l l ^ / . i i l > | ik/.S I i -^ l i/. l l i >\\\UMl-e\\U\\l > \4fi\\l ~ 4fi\\l since the sets are strictly-2-associated with S\ with constant of strict-2-associatedness c. Hence > i ' l l / l l ! - -'11/112 " 8 //// 80 Lemma 4.6 Suppose E satisfies the hypotheses of the theorem and S is an open subset of G. Assume E' C E is strictly-2-associated with the open set S2 whose closure is contained in S. There is a finite set Fx = Fi(E,E',S,S2) so that whenever E" = {Xi}ieI C E satisfies XiXJ1 fi Fx if i ^ j, then E' U E" is strictly-2-associated with S. (Compare with Lemma 3.11.) Proof Choose S\ open with S2 C Si C Si c S. By Lemma 4.3 there is a constant c > 0 which is a constant of strict-2-associatedness for Si and each of the sets E' U {X}, X G F. By Lemma 4.5 obtain the finite set F = F(c,SuS) with the property that if {Ei}iei are strictly-2-associated with Si with constant of strict-2-associatedness c, and EiEj1 n F = 0, then U,e/ Ei 18 strictly-2-associated with S. Since E has the uniformly large gap property the proof may be completed as was Lemma 3.11. / / / / Proof of Theorem 4.1 Since E is assumed to belong to Mn for some n we carry out an induction proof similar to that given for Theorem 3.5. The induction assumption will be as follows: Assume that for each non-empty, open subset S of G, whenever E' C E is strictly-2-associated with the open set Sx 81 whose closure is contained in S, and E" C E belongs to class Mk, then E' U E" is strictly-2-associated with S, Since the proof is very similar to the proof of Theorem 3.5 we present the details only for k = 0. Let S be any open, non-empty subset of G„ Let E' C E be strictly-2-associated with the open set Si, Si C S, and let E" C E belong to class M0. Choose S2 open with Si C S2 C S2 C S. Choose the finite set Fi = FX(E, E', S2, Si) from Lemma 4.6. Since E" tends to infinity there is a finite set F so that if X, rp G E" \ F and X ^  rp, then Xrp~l <£ Fx. By Lemma 4.6 E' U (E" \ F) is strictly-2-associated with S2. By Corollary 4.4 E' U B" is strictly-2-associated with S. //// Corollary 4.7 If i? satisfies the hypotheses of Theorem 4.1 and f e L%(G) vanishes on an open set, then / = 0. As with Theorem 3.5 the XQ-subtransversal!ty condition is necessary to obtain the corollary above. In [14] Mandelbrojt proved Theorem 4.8 Let E = {nk} C Z and let (3 = sup {a : £ diverges }. If 82 P < 1, f e L2E{T) and um Wp h(- hf''M')>JL, a^o+ - I n " 1-/? then f = 0. In particular, if / vanisies on an open set, then f = 0. If does not contain parallelepipeds of arbitrarily large dimension it follows from Corollary 2.18 that p < 1. Thus the conclusion of Corollary 4.7 can be obtained for such sets by Mandelbrojt's theorem. If however E is only assumed to satisfy the assumption of Mandelbrojt's the-orem, then E need not be strictly-2-associated with all open subsets of T. To prove this, observe that there are sets which satisfy Mandelbrojt's hypothesis and yet contain arbitrarily long sequences of consecutive integers. Such sets cannot be strictly-2-associated with all open sets, as a result of the next proposition. Proposition 4.9 Suppose E C Z contains arbitrarily long arithmetic progres-sions {<Jjv + 6,... ,ajf + Nb}, for some fixed b. Then E is not s trie tly-2- associa ted with any open set that does not have full measure. Proof Let S be any open subset of T with measure less than one. Let /jy be the JV'th partial sum of the function x i - > lsc(bx). 83 Suppose l 5c(x) = J2n cneinx. Then fN(x)= £ cneinbx = e-ia»*e-«N+1»*gN(x) \n\<N where gn G LE(G) if we assume that the arithmetic progression {aN + 6,..., aN + (2N + 1)6} C E . Now ||a iV(x)l5(6x)||2 = |l/J V(x)l5(6x)||2 and since {/N} converges to l<?c(6x) in L2(T), | |^(x)l5(6x) | | 2 —> 0. But \\9N\U = WfNh - l|l5C(6x)||2 = m(Sc)1/2 > 0. Thus E cannot be strictly-2-associated with S. //// It is possible to prove a similar statement for E C f although we do not give the details here. We do not know if the hypotheses that E belong to class Mn or have the uniformly large gap property are necessary for the strict-2-associatedness conclusion. As mentioned in Chapter two there are sets E in Z which tend to infinity but have arbitrarily long arithmetic progressions, thus E may be strictly-2-associated with all open subsets of G without being a A(p) set for any p > 0. Good necessary conditions for the conclusion of Theorem 4.1 are unknown at this time. 84 Open Problems We conclude by presenting a short list of open problems suggested by topics discussed in the thesis. 1. Are there sets which do not contain parallelepipeds of arbitrarily large dimen-sion and which are not A(4) sets? which are not A(p) sets for any p > 0? Lemma 2.27 together with [21, 4.5] can be used to show that sets which do not contain parallelepipeds of dimension 2 are indeed A(4) sets. If the answer to the first question posed above is no, then all A(p) sets, p > 0, are A(4) sets. If just the second part has a negative answer the union problem for A(p) sets would be solved. 2. If E C Z is a A(2) set, is E strictly-2-associated with all subsets of T with positive measure? 3. What conditions must E C F satisfy if (i) E is strictly-2-associated with all open, non-empty subsets of GI or (ii) no L\(G) function may vanish on an open, non-empty subset of G without being identically zero? 4. Are the results of this thesis true in the setting of A(p) sets in discrete, non-abelian groups? 85 Bibliography [1] Bachelis, G. and Ebenstein, S., On A{p) sets, Pacific J . Math, 54 (1974), 35-38. [2] Blei, R. C., Multidimensional extensions of the Grothendieck inequality, Arkiv for Mat., 17 (1979), 51-68. [3] Bonami, A., fttude des coefficients de Fourier des fonctions de LP{G), Ann. Inst. Fourier (Grenoble), 20, fasc. 2 (1970), 335-402.. [4] Bourgain, J. , Proprietes de decomposition pour les ensembles de Sidon, Bull. Soc. Math. France, 111 (1983), 421-428. [5] Dechamps-Gondim, M., Ensembles de Sidon topologiques, Ann. Inst. Fourier (Grenoble), 22, fasc. 3 (1972), 51-79. [6] Ebenstein, S., A(p) sets and Sidon sets, Proc. Amer. Math. Soc, 36 (1972), 619-620. [7] Edwards, R. E. , Hewitt, E. and Ross, K., Lacunarity for compact groups, I, Indiana J. Math., 21 (1972), 787-806. [8] Fournier, J. J. F., Two observations about 2-associatedness, preprint, (1982). [9] , Uniformizable A(2) sets and uniform integrability, Colloq. Math., to appear. [10] Fournier, J. J. F. and Pigno, L., Analytic and arithmetic properties of thin sets, Pacific J. Math., 105 (1983), 115-141. [11] Hajela, D., Construction techniques for some thin sets in duals of compact abelian groups, Ph. D. Dissertation, Ohio State University, (1983). [12] L6pez, J. , Fatou-Zygmund properties on groups, Ph. D. Dissertation, University of Oregon, (1975). [13] Lopez, J. and Ross, K., Sidon Sets, Lecture notes in Pure and Applied Mathe-matics, 13, Marcel Dekker, Inc., New York, 1975. 86 [14] Mandelbrojt, S., Series de Fourier et classes quasi-analytiques de fonctions, Gauthier-Villars, Paris, 1935, [15] Miheev, I. M., On lacunary series, Math USSR Sbornik, 27 (1975), 481-502; translated from Mat. Sbornik 98, 140 (1975), 538-563. [16] , Trigonometric series with gaps, Anal. Math., 9 (1983), 43-55. [17] Neugebauer, C. J. , Some properties of Fourier series with gaps, Contemporary Mathematics, 42 (1985), 169-174. [18] Pisier, G., Ensembles de Sidon et processus gaussiens, C. R. Acad. Sc. Paris, 286 (1978), 671-674. [19] Rudin, W., Fourier analysis on groups, Interscience Publishers, New York, 1962. [20] , Function theory in polydiscs, W. A. Benjamin Inc., New York, 1969. [21] , Trigonometric series with gaps, J. Math. Mech., 9 (1960), 203-338. [22] Szemeredi, E. , On sets of integers containing no k elements in arithmetic pro-gressions, Acta. Arith., 27 (1975), 199-245. [23] van der Waerden, B. L., How the proof of BaudeVs conjecture was found, Studies in Pure Math., presented to R. Rado, L. Mirsky, ed., London, 1971, 251-260. [24] Zygmund, A., On a theorem of Hadamard, Ann. Soc. Polon, Math., 21 (1948), 52-69; Errata, 357-358. [25] , Trigonometric Series, Volume I, Cambridge University Press, Cam-bridge, 1959. 87 

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