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Algebraic numbers and harmonic analysis in the p-series case Aubertin, Bruce Lyndon 1986

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ALGEBRAIC NUMBERS AND HARMONIC ANALYSIS IN THE p-SERIES CASE By BRUCE LYNDON AUBERTIN B.Sc., Massey University, 1970 M.Sc.(Hons), University of Auckland, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1986 © Bruce Lyndon Aubertin, 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of M /TTM £ M / f T l CS The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 i i Thesis advisor: Professor J.J.F. Fournier ABSTRACT CO For the case of compact groups G = II_. = ^Z(p)_. which are d i r e c t pro-ducts of countably many copies of a c y c l i c group of prime order p, links are established between the theories of uniqueness and s p e c t r a l synthesis on the one hand, and the theory of algebraic numbers on the other, s i m i l a r to the well-known results of Salem, Meyer et a l on the c i r c l e . Let p > 2 be a prime and l e t k{x }^ denote the p-series f i e l d of formal Laurent series z = l}1 a with c o e f f i c i e n t s i n the f i e l d J=-°° j k = {0,1,...,p-l}, and the integer h a r b i t r a r y . Let L(z) = - 0 0 i f a = 0 for a l l j ; otherwise l e t L(z) be the largest index h for which * 0. We examine compact sets of the form CO E(0,I) = {E. = 1e ie : e. £ 1} where 0 6 k{x - 1}, L(9) > 0, and I i s a f i n i t e subset of k[x]. If 0 i s a Pisot or Salem element of k{x - 1}, then E(8,I) i s always a set of strong synthesis. In the case that 0 i s a Pisot element, more can be proved, including a version of Bochner's property leading to a sharper statement of synthesis, provided certain assumptions are made on I (e.g., I z> { 0 , l , x , . . . , x L ( e ) - 1 } ) . Let G be the compact subgroup of k{x }^ given by G = {z: L(z) < 0}. Let 0 £ k{x - 1}, L(9) > 0, and suppose L(0) > 1 i f p = 3 and L(0) > 2 i f p = 2. Let I = {0,1.x,...,x 2 L ( 6 ) - 1}. Then E = 8 _ 1 E ( 9 , I ) i s a perfect subset of G of Haar measure 0, and E i s a set of unique-ness for G precisely when 0 i s a Pisot or Salem element. i i i Some byways are explored along the way. The exact analogue of Rajchman's theorem on the c i r c l e , concerning the formal m u l t i p l i c a t i o n of series, i s obtained; this i s new, even for p = 2. Other examples are given of perfect sets of uniqueness, of sets s a t i s f y i n g the Herz c r i t -erion for synthesis, and sets of m u l t i p l i c i t y , including a class of M-sets of measure 0 defined v i a Riesz products which are res i d u a l i n G. In addition, a class of perfect M^-sets of measure 0 i s introduced with the purpose of s e t t l i n g a question l e f t open by W.R. Wade and K. Yoneda, Uniqueness and quasi-measures on the group of integers of a p-series f i e l d , Proc. A.M.S. 84 (1982), 202-206. They showed that i f S i s a character series on G with the property that some subsequence {SpNj} of the p^-th p a r t i a l sums i s everywhere pointwise bounded on G, then S must be the zero series i f S N. -> 0 a.e.. We obtain a strong P J & complement to this r e s u l t by establishing that series S on G exi s t for which S 0 everywhere outside a perfect set of measure 0, and for n which sup |s N| becomes unbounded a r b i t r a r i l y slowly. i v TABLE OF CONTENTS Abstract 1 1 L i s t of symbols vi Acknowledgements v i i INTRODUCTION 1 CHAPTER I THE p-SERIES FIELDS - ILLUSTRATIVE AND PRELIMINARY RESULTS 1. The p-series f i e l d s 8 2. Two examples of analogy with the r e a l numbers - the prime number theorem for k[x] and the uniform d i s t r i b u t i o n mod 1 of c e r t a i n sequences 11 3. The set T of Pisot and Salem elements 21 4. Harmonious sets and coherent sets of frequencies . . . . 23 5. The connection with members of T 26 6. Character series on the compact group G - and some pro b a b i l i t y considerations . . . . . 32 7. Formal m u l t i p l i c a t i o n of series - and the analogue of Rajchman's theorem on the c i r c l e 37 CHAPTER II UNIQUENESS OF REPRESENTATION 1. The problem of uniqueness . . . . . 48 2. Classes of perfect U-sets - and discussion of a general problem 60 3. M-sets of measure 0 v i a Riesz products 74 4. A class of perfect Mg-sets of measure 0 77 5. N u l l series with slow growth - and a theorem of Wade and Yoneda gi 6. Uniqueness and members of T - results for sets E(0,I) 35 7. Some comments, with a question or two 95 CHAPTER III SPECTRAL SYNTHESIS 1. The problem of synthesis 99 2. Results for sets E(9,I) 103 V 3. Exercising l i g h t l y on the t r a i l s of Herz and Meyer . . . 108 4. Bochner's property when 9 i s a Pisot element and the proof of Theorem 11 113 BIBLIOGRAPHY 123 v i LIST OF SYMBOLS In most cases our notation i s very standard and we do not l i s t , oo for example, L (G). Nevertheless, we think the following l i s t may be found h e l p f u l . p a prime, fixed throughout this thesis, but used on occasion for other things as well k the p-element f i e l d {0,1,...,p-l}, but also used often as a dummy index k[x] the ring of polynomials in the indeterminate x with c o e f f i c i e n t s i n k k(x) the f i e l d of quotients k{x }^ the p-series f i e l d K the algebraic closure of k{x ^}, though possibly just some compact set or other T the set of Pisot and Salem elements i n k{x - 1} u,z,... usually members of k{x _ x} [z] and (z) the i n t e g r a l and f r a c t i o n a l parts of z 0)Q,O)^ ,(^2, • • • standard l i s t i n g of the members of k[x] G,r,r ,... always groups, t h e i r precise meanings subject to change (with due notice) X a character, not necessarily continuous Y always a continuous character e^  maps k{x }^ to [0 , °° ) and enumerates k[x]: e.( w n) - n L the degree function on k{x H PM a space of pseudomeasures R the r e a l numbers a.a. almost a l l v i i ACKNOWLEDGEMENTS The author i s indebted to Professor John Fournier for his encouragement and support during the production of this work. He is' also indebted to Professor D.W. Boyd, who, with his expertise on number theory, checked through much of this thesis. A spe c i a l debt of gratitude i s owed Professor John Coury, and not only for his moral support, but for showing us the charms of the Walsh system i n the f i r s t place. Some of the research work for this thesis was carried out i n the Department of Mathematics and S t a t i s t i c s at Massey University, Palmerston North, New Zealand, and the author would l i k e to express his appreciation of the generous support he received while he was there. He also wishes to thank The University of B r i t i s h Columbia for i t s generous f i n a n c i a l assistance. F i n a l l y Marie, Anne, Bobbie, Ross, Lynne, Amy and Bob. And my friends Drs Ersan and Yilmaz A k y i l d i z . It i s hard to imagine how this thesis could have been written without you. g i i i for Gudrun and Robert and for B r i g i t t e 1 INTRODUCTION The Rademacher functions [30] are defined by +1 0 <_ x < h -1 h < x < 1 extended with period 1 to the whole of R, and r n ( x ) = r Q ( 2 n x ) , n = 1,2,3, They form a sequence of i . i . d . (independent and i d e n t i c a l l y distributed) random variables on the unit i n t e r v a l , and an orthogonal system of 2 functions there (for the space L of square-integrable functions). But they are not complete. Their completion was i d e n t i f i e d by Walsh [42] as the set of a l l f i n i t e products of these functions (this had been effected also by Rademacher himself, i n an unpublished work of 1921) . Paley too [27] studied these functions, and the Walsh system, i n Paley's enum-eration, i s the complete orthonormal set of functions given by w Q(x) 5 1 and w (x) = r (x)...r (x) for n = 2 n i + . .. + _2 nv , where ^ > ... > n v _> 0. Thus every integrable function f on the i n t e r v a l has a Walsh -i Fourier series f(x) ~ Z „ c w (x), where c = J f(x)w (x)dx. n=0 n n n J0 ' n v ' Early authors, notably Paley [27], studied the Fourier properties of the Walsh system. Further systematic investigations were undertaken, r Q ( x ) - { 2 concurrently and independently, by Fine [12] and Vilenkin [37]. Vilenkin looked at the system i n a much broader context, while Fine's work was much closer to the t r a d i t i o n of the c l a s s i c treatises on trigonometric series by Bary [ 2 ] and Zygmund [46]. There are remarkable s i m i l a r i t i e s with the trigonometric system. There are some s t r i k i n g differences too. I t i s w e l l known, for example, that the Fourier c o e f f i c i e n t s of an absolutely continuous function f decay at a rate which i s o ( l / n ) . But as Fine showed, i f the Walsh -Fourier c o e f f i c i e n t s of such an f decay at this rate, then f must be constant! ..- Fine considerably strengthened the analogy with the trigonometric system when he showed that the Walsh functions w n ( x ) are e s s e n t i a l l y the continuous characters (complex homomorphisms into the unit c i r c l e ) oo on the dyadic group G = IL_^Z ( 2 ) _ . , just as the trigonometric functions exp(2TTinx) are the continuous characters on the c i r c l e group (here, the c i r c l e i s taken to be the i n t e r v a l [0,1), with addition modulo 1, and the dyadic group G i s simply the group of a l l sequences of 0's and l ' s , with coordinate-wise addition modulo 2 ) . Later, Fine [13] introduced the dyadic f i e l d (the 2-series f i e l d ) ; the dyadic group f i t s i n here ju s t as the c i r c l e f i t s i n with R. There i s much current i n t e r e s t i n the Walsh system. Two survey a r t i c l e s , [1] and [38], have been written on the topic of Walsh series; look too for the forthcoming book [40] by Wade et a l . The applications of c l a s s i c a l Fourier analysis, both within and outside of mathematics, are legion. Does the Walsh system find such applications too? It does, and i n many areas the Walsh transform i s just the right transform to 3 use. Applications to physical problems, in e l e c t r i c a l engineering, information theory and so on, are numerous (some pointers to the l i t e r -ature here w i l l surely be found i n the bibliography of [40], but l e t us mention the books [19] and [27]). Now l e t us change the subject a l i t t l e b i t , and focus on one of the incredible discoveries of Salem, who f i r s t made the Pisot numbers1 famous in harmonic analysis. A subset E of the unit c i r c l e i s c a l l e d a set of uniqueness (or U-set) for trigonometric s e r i e s , i f the zero series i s oo 2TTinx the only series of the form E ^ a e which can converge to 0 pointwise outside E. For a r e a l number 8 > 2, l e t C(9) be the Cantor subset of the c i r c l e defined by C(9) = { E ^ e . ^ - 1 : e = 0 or 1}. The set C(3) i s a simple d i l a t i o n of Cantor's c l a s s i c a l ternary set (indeed, 2C(3) ±s^ the ternary set on [0,1]). Now these sets C(G) are a l l perfect sets of measure 0, and i n many respects they are a l l remarkably the same. But C(0) i s a U-set i f and only i f 0 i s a Pisot number! The Pisot numbers are part of the class T of algebraic integers 0 > 1, a l l of whose conjugates (other than i t s e l f ) l i e inside the closed unit disk |z| _< 1. They are the ones whose conjugates are a l l s t r i c t l y inside the disk |z| < 1. The remaining numbers i n T are known as Salem numbers, and they are intimately related to the Pisot numbers. (The sets of Pisot and Salem numbers on the l i n e are usually denoted by S and T respectively; however, since i t i s the union of these, two sets, i n the p-series context, that plays a central role i n much of the l i . e . , the Pisot-Vijayaraghavan numbers; some authors use the alternat-ive contraction 'PV number' for these numbers. 4 present work, we have opted to denote this union by the single l e t t e r 'T'.) We had a 'thesis', and i t was that this b e a u t i f u l theorem of Salem's could not f a i l to have an analogue for Walsh series. Sneider [35] wrote one of the early papers on the uniqueness theory for Walsh series; he proved a version of Rajchman's theorem, and showed that the sets C(2 ), k >_ 2, are a l l Walsh U-sets. To this day, i t i s not known whether any of the other sets C(9) are (or are not) U-sets for Walsh series (see also section 7 i n chapter II of the present work). Wade [39] had prepared some of the foundations for an ana Logi.ie of Salem's 2 r e s u l t . Alas, Yoneda [43] h.id then shown that a l l of the sets r 0 0 * " i t F(9) = U i = 1 £ i 9 : E. = 0 or 1} constructed i n the dyadic f i e l d are U-sets for Walsh series (here 6 must be big enough to make F(9) a set of measure 0). No progress was made u n t i l by chance, i n chasing down the prime number theorem for k[x] (which i s n ' t ours!), and i n exam-ining certain questions of uniform d i s t r i b u t i o n of sequences, we came upon the papers of Bateman and Duquette [3] and Grandet-Hugot [15], [16] on the remarkable sets T of Pisot and Salem elements i n f i e l d s of formal power series. We owe a l o t to these authors, and to the excellent b i b l i o -graphy i n [25] on the topic of uniform d i s t r i b u t i o n of sequences! Now l e t us change the subject s l i g h t l y once again. Meyer (see [28]) showed that i f 9 i s a Pisot number, then the compact set C(9) obeys the following strong form of spectral synthesis. For every bounded continuous 2 The result we are r e f e r r i n g to as Salem's, namely that C(8) i s a U-set for the c i r c l e precisely when 0 i s a Pisot number, was proved by Zygmund and Salem i n 1955. 5 function (j) on R, whose spectrum l i e s i n C(8) , there i s a sequence (})^ , k ^ 1, of f i n i t e trigonometric sums whose frequencies belong to C(8), such that supltj), I < sup I (j) I, and <J>, <|> uniformly on compact subsets of R. R R k Herz [21] had proved this for the case of the ternary set (indeed for the case of C(9) for a l l r a t i o n a l integers 8 > 1). The result for a l l Pisot numbers had been a conjecture of Salem's, and was proved by Meyer using a generalization of the basic d i s c r e t i z a t i o n technique employed by Herz for the ternary set (enter Meyer's harmonious sets ) . I t i s not known i f any of the sets C(8) obey (or disobey) synthesis when 8 i s not a Pisot number. Our results on synthesis for the p-series case are based very much on the methods of [28], which forms the standard reference for much of this thesis. On the other hand, the proof of our analogue of Salem's uniqueness result i s i n a more c l a s s i c a l vein, much as i s found i n Salem's elegant l i t t l e book [32] (Meyer gives Salem's result a new treatment. We too, however, bypass the c l a s s i c a l route, i . e . , the one through Minkowski's theorem on complex l i n e a r forms). These two famous problems i n harmonic analysis, of characterizing the U-sets and the sets obeying synthesis, are subtle ones indeed, and the results of this thesis show they are l i k e l y to be every b i t as delicate i n the Walsh case (or p-series case) as they are on the c i r c l e and r e a l l i n e . We note that the limelight which i s taken by the Pisot numbers i n the c l a s s i c a l results on uniqueness and spectral synthesis for the sets C(0), seems, in the setting of a p-series f i e l d , to be shared by a l l members of T. There does appear to be a turn around, how-ever, i n chapter I I I , where synthesis i s obtained for a l l members of T, 6 but Bochner's property only for the Pisot elements. Meyer [281 gets synthesis only for the Pisot numbers, but has Bochner's property for a l l members of T ( i f the reader finds this a l i t t l e confusing, we promise to try to be less c r y p t i c i n the main body of the t e x t ! ) . Analogues of the c l a s s i c a l uniqueness and synthesis results for the sets C(8) are known i n the f i e l d s of p-adic numbers; these can be found i n [28]. There have been many contributors to the work surrounding and evolved from Salem's, and the reader should consult more authori-tative sources (such as [28]) for precise references and c r e d i t s . Some of what we do here could have been done for certain Vilenkin groups of bounded type (groups G = IL = 1 Z ( p j ) , for {p} a bounded sequence of primes). For example, the exact analogue of Rajchman's theorem holds there, and the construction of perfect M^-sets i n section 4 of chapter II could have been carried out there as w e l l . We have decided however, for the sake of greater c l a r i t y and unity, to stay with the p-series case throughout. Appended to some, but not a l l , of the sections i n this thesis there w i l l be Notes, where further comments and/or references w i l l be given. This w i l l help streamline the main text, although comments and references w i l l be found there as well. Some of the sections w i l l have subsections, and we w i l l make references as in the following examples: §11.6 denotes section 6 i n chapter I I , §11.6.3 subsection 3 of §11.6, and so on. Displays w i l l be numbered consecutively (1), (2),... as w i l l be lemmas, theorems and so on (separately)'. A mini-index w i l l be found at the beginning of some sections for the subsections which they contain. 7 S t i l l one t a n t a l i z i n g question does remain (among the many others). Is the ternary set a U-set for Walsh s e r i e s on the unit i n t e r v a l , j u s t as i t i s for trigonometric series? 8 CHAPTER I THE p-SERIES FIELDS - ILLUSTRATIVE AND PRELIMINARY RESULTS There are remarkable s i m i l a r i t i e s between the f i e l d of r e a l numbers and certain f i e l d s of formal power series, and between the corresponding trigonometric systems on them. This i s ref l e c t e d too i n their corresponding 'number theories', and i n this chapter, as we lay some of the groundwork for the l a t e r chapters, we w i l l take the time to watch some of the interplays, involving both number theory and the theory of p r o b a b i l i t y , that go on i n the p-series case. §1. The p-series f i e l d s k{x Let p be a prime :> 2, and l e t k denote the p-element f i e l d {0,1,...,p-l}. k[x] w i l l denote the ring of polynomials i n the indeter-minate x with c o e f f i c i e n t s i n k, and k(x) i t s f i e l d of quotients. The p-series f i e l d k{x i s the f i e l d of formal series of the form /1\ h , h-1 . vh j (.1) z = a,x + a, ,x + ... = l. a.x h h-1 J= - 0 0 j where the integer h i s not fixed, and the c o e f f i c i e n t s are a l l i n k. Define the degree function L on k{x }^ by L(z) = h for z as i n (1) with a, * 0, and define L(0) = -°° . n The absolute value of z i s defined by (2) I zI = p when L(z) = h. 9 Elements i n k(x) can formally be expanded in Laurent series of the form (1), and k{x - 1} i s simply the completion of k(x) with respect to the absolute value defined by (2). This absolute value has a unique extension to the algebraic closure K of k{x where i t s a t i s f i e s + z^l £ max{Iz^I , IZ2 I } • The absolute value on k{x }^ gives r i s e to a metric topology which turns k{x }^ into a l o c a l l y compact t o t a l l y disconnected f i e l d . This topology has a base at 0 consisting of the compact open subgroups G R = {z £ k{x _ 1}: L(z) < -r} *"* IT of (normalized) Haar measure m(G^) = p , with the. compact group playing a p a r t i c u l a r l y distinguished r o l e . Define Y Q ( Z ) = 1, and the fundamental character on k{x }^ by Y 1(z) = exp(2iTia_ 1/p) , z = ) : j = _ ro a x^ . For z, w € k{x define Y z by Y z(w) = Yj_(zw) . Then the mapping z ->• Y z i d e n t i f i e s k{x }^ with i t s dual as an additive group, and k[x] with the dual group of G Q . Every z 6 k{x }^ can be written uniquely i n the form z = [z] + (z), [z] £ k[x], (z) e G 0 , which s p l i t s z into i n t e g r a l and f r a c t i o n a l parts.^ The mapping between k{x and the reals given by 10 (3) e a .x' J J J i s a continuous, measure-preserving, e s s e n t i a l l y one-to-one map of k{x } onto [O,00) which carries G n onto the unit i n t e r v a l [0,1], and k[x] provides us with a natural ordering {YQ»Y-^>...} of the dual group of G Q, and indeed a natural ordering of k[x] i t s e l f . In the sequel, we s h a l l index characters y on GQ by either members of k[x] or nonnegative integers, whichever i s convenient at the time. When p = 2, G Q i s the dyadic group introduced by Fine [12], and the functions {Y n: n = 0,1,2,...} are the Walsh functions i n Paley's enumeration. The f i e l d k{x }^ i n this case i s known also as the dyadic  f i e l d , and the mapping (3) plays an important role i n Walsh analysis on the r e a l l i n e , enabling as i t does a transfer of many results between k{x and the rea l s . Note that under (3), the subgroup G ^ , for each integer r, maps to the r e a l i n t e r v a l [0,p r ] , and cosets of map to intervals of the form [jp ,(j+l)p ] where j i s a nonnegative integer. The c h a r a c t e r i s t i c function of any of the subgroups G i s a f i n i t e character sum, and with ( G R ) being a neighbourhood base at 0 for the topology, this i s a nice thing to have indeed. For nonnegative i n t e g r a l r, we have from which i t i s seen that the c h a r a c t e r i s t i c function of any coset of i n j e c t i v e l y onto the nonnegative integers. This enumeration of k[x] (4) such a G has also the form £ r 11 Notes. An account of p-series f i e l d s , where the perspective i s a l i t t l e d i f f e r e n t from ours, can be found i n Taibleson [36]. The compact group i s the unique maximal i d e a l i n the ring of integers of k{x 1} (cf. [36]). Our notation i s taken from [3]. In much of the exist i n g l i t e r a t u r e on work related to this thesis (e.g., [39]), the p-series f i e l d s are not regarded e x p l i c i t l y as f i e l d s of series; they have already been mapped as in formula (3) to [0,°°), and ars dealt with there using a d i f f e r e n t notation. We note that,- for most purposes, the absolute value defined i n (2) could just as well have been given by l z | = c for any fixed constant c > 1, We have chosen c = p so that we could write (5) • / . f(az)dz = lal L J . f(z)dz k ( x - 1 } k{x - 1} for functions f £ L^(k{x *}) and nonzero a £ k{x (the integration i s with respect to Haar measure). In f a c t we w i l l have no occasion to use (5) e x p l i c i t l y ( i t s use i s i m p l i c i t , however, i n parts of chapter III where we deal with d i l a t i o n s and spectra of functions). §2. Two examples of analogy with the r e a l numbers - the prime number ' theorem for k[x], and the uniform d i s t r i b u t i o n mod I of certain sequences. N.B. No great o r i g i n a l i t y i s claimed for the results i n this section; we 12 wish only to entertain the reader. Proposition 1 below (§2.1) should do this when we state the already known prime number theorem for k[x] i n perhaps a new way (do not look here for a stunning new proof of i t from scratch!). Proposition 2 i n §2.2 (which begins page 17) i s not a l l that s u r p r i s i n g , and i s probably well known. What we wil] do there i s give i t a novel proof. §2.1 The prime number theorem for k[x]. How many prime numbers are there < n ? Let TT(n) be this number, and r e c a l l the famous prime number theorem : By a prime i n k[x], we mean an i r r e d u c i b l e polynomial there of degree at least 1. Very general prime number theorems are available i n abstract settings [24]. When i t comes to rings of polynomials over a f i n i t e f i e l d , very precise calculations and estimations are possible. Let e^  be the natural enumeration of k[x] given by (3); i . e . , e^  i s the 1-1 onto mapping of k[x] onto the nonnegative integers where TT(n) n/log n, n ->• °°. e(co) = n for CO vm i " V i x £ k[x] and E^a.p 1 = n. Let P be the set of primes i n k[x], and l e t TT (n) = #{co € P: e(w) < n} P - -where it denotes the c a r d i n a l i t y of a set. PROPOSITION 1. n/log n, n ->- <». 13 Proposition 1 i s actually very easy to obtain for n skipping to °° along a sequence n = p m (m °°) . In t h i s , the ir r e d u c i b l e s are counted according to their degree (see Prop. 2.3 on p. 60 of [23]). As already mentioned, some authors prefer to do their p-series arithmetic on [0,°°). Let us agree to c a l l a natural number n a Walsh prime i f e^ (to) = n for an i r r e d u c i b l e co £ k[x], with k = {0,1} ( i . e . p = 2). Let '•' denote 2-series m u l t i p l i c a t i o n of natural 2 2 numbers. Now ( 1 + x ) = 1 + x as many a high school graduate w i l l 2 know. We have e_(l + x) = 3 and e ^ l + x ) = 5, and so 3*3 = 5. Thus 5 i s not a Walsh prime. The f i r s t few Walsh primes are 2, 3, 7, 11, 13, 19, 25, 31, ... (we could go on, with our l i s t of 1500 of them prepared with the assistance of P. Borwein!). Proposition 1 t e l l s us that the number (n) of Walsh primes <^  n obeys ^ ( n ) ~ n/log2n as n -»• 0 0. To get Proposition 1, which we do not think i s commonly known i n this very pretty form, we star t with a gleaning from Th. 1.3 on p. 102 of Hayes [20]. Let s > 0 be fixed, and r >_ s + 1. Let a, a^, , • • • , a g be members of our p-element f i e l d k, with a * 0. Then //{to £ P: co begins ax r + a^x r * + ... + a s x r S + . . . } (6) = p r " S / r + 0 ( p r / 2 / r ) . Now the reader may think Proposition 1 i s obvious from here, but i n any case we s h a l l record our j u s t i f i c a t i o n for i t . The implied constants i n our big O's below may not always be the same, but we w i l l end up with one global big 0 to y i e l d Proposition 1, which i s a l l we want (so our computations may be a l i t t l e rough and ready). 14 Choose s = 0 in (6) and sum over a =t= 0 to get (7) #{a> 6 P: deg(w) = r} = ( p - l ) p r / r + 0 ( P r / 2 / r ) , then sum over r < m to get (8) #{u £ P: deg(u) < m} = pm/m + 0( P m/m2). Now f i x s > 0 again. Fix b * 0, b. , b„, b a r b i t r a r i l y — i I s ( a l l members of k). For arbitrary m :> s + 1, l e t w* = bx m + b^x m *+...+ b g x m S . Now k[x] has the ordering given by £ , so co < p for example, i f and only i f e(a)) < _e(p) . We have //{u> € P: eo < co*} = / / ( ( D £ P: deg(co) < m} (9) + I G P: 0) begins ax™ + ...} 0<a<b + Z € P: a) begins bx m+ a.x1"" + ... + a x 1 s (a . . .,a ) I s m-1, , m-s^ , m—1, ,, m—s a^p +..>+asP < b^p +..'+bgp T l + T2 + T 3 * s a y -Now Tj = pm/m + 0(p m/m 2) by (8), and T 2 = (b-l)p m/m + 0(p m / 2/m) by (7) also, each term i n = p™ S/m + 0(p m /' 2/m), by (6). s-1 s-2 How many terms are there i n ? Answer: b^p + b^p + ... + 1 15 Adding everything up, we get for (9) that #{OJ G P: co < co*} = bpm/m + ( b l P S _ 1 + ... + b s)p m~ S/m + 0 ( P m / m 2 ) (10) = e(co*) + 0(p m/m 2). Let n = e(co ) ; then (10) gives y n ) 1 ° V = ^ . l o g ^ + log^n 0 ( pm / m 2 ) n m n n (11) = 1 + 0(l)/m + 0(l/m) 1 + 0 ( l / l o g n) . (Note that n = bpm'+ b^p m ^+ ... + b s p m S , and log n = ra + 0(1).) This gives Proposition 1 for n -* °° along a subsequence of the type n = bp + b^p + ... + b gp , m -> oo , for each fixed s . To get Proposition 1 for n ->- °° a r b i t r a r i l y , l e t e > 0, and choose s > 0 large enough that e(to*) < (1 + e) e(u)*) — — s for a l l large enough to , where to i s determined from to according to * , m , , m—1 , , , m—s , to* = bx + b.x + ... + b x +..., <12> * i S u>* = bx m + b l X m _ 1 + ... + b x m _ S . s i s (here we see e(to*)/e(to*) < 1 + p m _ S / p m = 1 + p" s) . Now l e t co* be given a r b i t r a r i l y as in (12) and l e t n = e/to*) , and 16 n = e(u)*) . We have s — s {to € P: to < 0)*} c {to £ P: to < to*} s c {u £ P: u < to*} U {co e P: OJ = to*}, s s s from which we get (13) TT (n ) < 7r (n) < IT (n ) + #{UJ € P: to = oo*}. p s — p — p s s s From (6), there i s a K = K(s) such that #{to £ P: w„ = to*} < pm~S/m + K p m / 2/m . Now (13) gives s s — 7 TP( n s ) m < V n ) - < W - + p _ s + o(p" m / 2>> (l+e)n s so l e t t i n g m ->• °°, we get for our fixed s , / i , „-v-l - • . £ TT (n)log n , . IT (n) log n , , (1 + £) < lim m f p ___£_ < lim sup p j£ 1 + n ->- °° n n->°° n Since e and s were a r b i t r a r y , the proof of Proposition 1 i s complete. 17 § 2 . 2 Uniform d i s t r i b u t i o n mod 1 of certain sequences. We recommend [25] for a wealth of information on the fascinating topic of uniform d i s t r i b u t i o n of sequences. Recall that a sequence {x J- , of r e a l numbers i s called uniformly distributed modulo 1 (written n n=I * u.d. mod 1) i f each subinterval I = (a,b) of the unit i n t e r v a l gets i t s proper share of the f r a c t i o n a l parts (x^) of the sequence: (1/N) Z^ = 1 X x(x ) m(I) = b-a as N -> °°, for a l l I = (a,b) c (0,1), where Xj denotes the c h a r a c t e r i s t i c function of I, extended to a l l of R with period 1. There i s the w e l l -known Weyl c r i t e r i o n for t h i s : {x } i s u.d. mod 1 i f and only i f n (14) (1/N) L N .exp(2Trimx ) •+ 0 n—1 n for a l l integers m * 0. The Weyl c r i t e r i o n shows very simply that _ CO (15) {nx} , i s u.d. mod 1 i f and only i f x i s i r r a t i o n a l , n=l J which i s a c l a s s i c a l result oE Weyl. Another more d i f f i c u l t c l a s s i c a l r e s u l t of Koksma i s the fact that {A9n} i s (i) f o r each X * 0, u.d. mod 1 for almost a l l 9 > 1; and ( i i ) for each 6 > 1, u.d. mod 1 for almost a l l X (see espe c i a l l y pages 32-40 of [25]). Let us spend a l i t t l e time i n the p-series case. A sequence z £ k{x i s cal l e d u.d. mod 1 i f each open subset A of the group 18 GQ = {z: L(z) < Of gets i t s proper share of the sequence of f r a c t i o n a l parts (z ) (in the sense described above). Again one has the Weyl n c r i t e r i o n (counterpart of (14)) for t h i s : lz } . i s u.d. mod 1 i f and only i f n n= l (16) (1/N) z l _ . r j z ) + 0 n — l m n for a l l integers m > 0 (or i f you prefer, for a l l nonzero members m of k[x] ) . Of course, ^ z n ^ ^ R u-d- m°d 1 i f and only i f i t s image under the mapping e_ i n (3) as a r e a l sequence i s u.d. mod 1 on R. This we leave as an exercise for the reader. Let us deal with the counterpart of (15) i n some d e t a i l . We remind the reader that whenever an ordering of k[x] i s t a c i t l y assumed, then i t i s the one given by the mapping e_ i n (3) of §1.1. Let us be e x p l i c i t here and l i s t the elements of k[x] as Uin, oo. , eo0, where co i s U 1 t. n defined by (17) e(w ) = n, n > 0. — n — Suppose z £ k(x), say z = r(x)/q(x) where r, q £ k[x]. We leave i t for the reader to see that the sequence (u^z) , n :> 0, of f r a c t i o n a l parts of the sequence ^ u n z ^ must be c y c l i c , with only f i n i t e l y many d i s t i n c t points, so cannot be u.d. mod 1. Now suppose z £ k{x ^}, but z € k(x), and l e t us apply the Weyl c r i t e r i o n (16) for u.d. mod 1 to the sequence i ^ z j ^ ^ . Choose an integer m > 0. Then (18) (1/N) < = 1 Y m ( V ) = (1/N) < = 1 Y n ( V ) • 19 Since z t k(x), the fixed point oo z cannot belong to k[x]. As i s the m case with the trigonometric system, i t i s well known that the D i r i c h l e t kernel n = 1 ^ i Y , N > 1 , i s uniformly bounded on compact subsets N n=0 n — of G Q which do not contain 0 . Thus the right member of ( 1 8 ) goes tc 0 l i k e 1 / N , so we have proved the counterpart to ( 1 5 ) (cf. [ 2 5 , p. 3 3 0 ] ) : (19) {oo z} i s u.d. mod 1 i f and only i f z 2 k(x) . n What can be said about the d i s t r i b u t i o n of the f r a c t i o n a l parts (oo z) for a subsequence n. < n_ < . . . ? Applying the Weyl c r i t e r i o n here, we get as i n (18), (20) (1/N) J ^ . Y ^ t o z) = (1/N) I1? ,Yn (w z) for m > 0. j = l m n j j = l nj m _ CO Now i f the sequence {n-s}._. i s s u f f i c i e n t l y lacunary (n ./n. > p for J 3~ i- 3+i- 3 a l l j j>. 1 i s enough), then one can show that the sequence ^ n . ^ - > i i s a sequence of i . i . d . random variables with mean 0. By the strong law of large numbers (see, for e.g., [10, p.126, Th.5.4.2]), the right side of (20) converges to 0 for almost a l l z i n G Q (the pro b a b i l i t y space). It follows that f o r a.a. z 6 k{x the e x p r e s s i o n Jr. (20) tends to 0 f o r everv m > 0. and t o r each z f o r which t h i s o c c u r s , we have that the sequence {oo z} i s u.d. mod 1. Now compare what we have just proved for the lacunary sequence { n^_} with what we have i n (19) for the f u l l sequence ^ u n ^ ' There i s an urge to interpolate. Return to (20) and suppose now n^ < < ... i s any sequence of po s i t i v e integers. The f i r s t thing one meets i n Chung's [10] section 5.4 on the strong law of large numbers i s an old lemma of Kronecker's: 20 Kronecker s lemma. Let i X j . L _ ^ be a sequence of complex numbers, and {a-;}. , a sequence of positive r e a l numbers increasing to °° . N If I. x./a. converges, then (1/a.J Z. . :x. -> 0 . 3 3 3 « J = 1 J 2 Now since X l / j ^ < 0 0, we can define a function f (z) i n L ( G Q ) by oo (21) f(z) = I (l/j)Y n.(co t nz) (for each fixed m), and from the p-series analogue of Carleson's theorem (see B i l l a r d [5]), we know that the right side of (21) converges to f(z) for a.a. z £ G Q (heavy machinery indeed!). For each z £ G Q for which we have this convergence, we can apply Kronecker's lemma to conclude that the right member of (20) tends to 0. Again i t follows that for a.a. z i n k{x ^}, (20) tends to 0 for a l l m > 0, and thus we have proved PROPOSITION 2. Let n^ < < ... be any increasing sequence of positive integers. Then the sequence {co z}. . i s u.d. mod 1 for almost a l l z £ k{x }. Let n-u denote the p-series m u l t i p l i c a t i o n of the natural number n with u £ [0,°°) (this i s unambiguously defined v i a (3) for a l l but a countable set of u ; as mentioned e a r l i e r , some authors work on [0,°°). See in p a r t i c u l a r , [39]). Then Proposition 2 can be stated i n r e a l terms as follows: Let 0 < n^ < < be a sequence of positive integers. Then the sequence {n.-u} . . i s u.d. mod 1 for almost aJ.1 u £ [0,°°). 21 Notes. Of course this i s the wrong proof of Proposition 2 (you may say), and what we have just presented i s a f a i t h f u l record of i t s 'discovery'. Now one finds i t s r e a l analogue as Th. 4.1 on p. 32 of [25], a r e s u l t of Weyl, with probably the 'right' proof. Naturally, the result there can be given a wrong proof similar to ours too. Notice we have not yet mentioned sequences {X9n} i n k{x ^}; r e s u l t s analogous to the r e a l case (see the li n e s following (15) on page 17) must surely be known for them, and we repeat that Proposition 2 i s probably not new. We are not f a m i l i a r with the l i t e r a t u r e on uniform d i s t r i b u t i o n i n a p-series context, and the reader should consult [25] (see espe c i a l l y pp. 328-330) for results and references here. In the next section, we s h a l l be interested i n certain 6 € k{x }^ for which nonzero A exist such that the f r a c t i o n a l parts (A9 n) are about as badly d i s t r i b u t e d i n G as they can be. ooOoo §3. The set T of Pisot and Salem elements i n k{x }. These elements are defined i n complete analogy with the r e a l Pisot and Salem numbers, with k[x] playing the part of the integers, k(x) the f i e l d of r a t i o n a l numbers, k{x }^ the r e a l f i e l d , and K the f i e l d of complex numbers. Thus we say an element 9 £ k{x }^ belongs to T i f 22 |0| > 1, and 0 i s a zero of a monic i r r e d u c i b l e polynomial of the form P(X) = X s + a.X S - 1 + ... + a 1 s where a^, ..., a £ k[x], and where the remaining roots 8 s of P(X) = 0 i n K (the conjugates of 0) have absolute values 1 8 J _< 1 for each i = 2, s. The Pisot elements are those members of T whose conjugates are a l l s t r i c t l y inside {z £ K: |z| < l } , and the Salem elements are the remaining members of T. Each member of k[x] having degree _> 1 i s p l a i n l y a Pisot element, but no other members of k(x) can belong to T. The set T w i l l be involved i n much of what we do i n this thesis; as mentioned e a r l i e r , the whole of T takes centre stage i n our uniqueness and spectral synthesis results i n a way taken only by the Pisot numbers i n the r e a l case. Let us note one contrast at this point. On the l i n e , the Pisot numbers form a count-able closed set, as was shown by Salem and i s well known. But in k{x elements of the countable set T (indeed the Pisot elements) are known to be everywhere dense i n the set {z £ k{x ^ } : I z I > 1} (see [16]). Bateman and Duquette [3] have characterized the Pisot elements i n k{x •'*} as follows: i f 8 £ H x - 1 } and 181 > 1, then 8 i s a Pisot element i f and only i f there exists A £ k{x ^}, A * 0, such that the f r a c t i o n a l parts ( A 8 n ) -»• 0 as n -> 0 0 . In the sequel, we s h a l l make use of the following characterization of members of T (see [15]). THEOREM (Grandet-Hugot). If 0 £ k{x - 1} and 101 > 1, then a nec-essary and s u f f i c i e n t condition for 8 to belong to T i s the existence of A £ k{x - 1}, A * 0, such that I ( A 0 n ) | < 10|" 2 for a l l s u f f i c i e n t l y large n. 23 Notes. Such a X s a t i s f y i n g the conditions i n either of the two characterizations just given necessarily belongs to the f i e l d of 0 ( i . e . , to the f i e l d k(x)(0)). Pisot elements i n k{x (for more general f i e l d s k ) were introduced i n [3] and studied further, along with the Salem elements, i n [15], [16]. Some high-precision results on the set T can be found also in [8], where the density r e s u l t of [16], mentioned above, has been complemented as follows: the Salem elements are dense i n each of the sets {z £ k{x *}: L(z) = s}, where the positive integer s i s r e l a t i v e l y prime to p . On the l i n e , as already mentioned, the set S of Pisot numbers i s closed, but i t i s not known i f the set of Salem numbers i s closed. A famous Pisot number on the l i n e i s 0* = (l+/5)/2 , the golden r a t i o of the Greeks. 8* i s the smallest l i m i t point of the set S . References for this and some related results can be found i n [6]. The Pisot and Salem elements i n a p-series f i e l d c e r t ainly appear to be no less i n t e r -esting or remarkable than t h e i r cousins on the l i n e ; see [28] for more results and references concerning the r e a l case. **** §4. Harmonious sets and coherent sets of frequencies i n k{x ^}. If T i s a I.e.a. ( l o c a l l y compact abelian) group, one would l i k e i d e a l l y to have discrete subgroups D for the purpose of atomizing d i s t -ributions on r . Where such discrete subgroups are lacking, harmonious sets of frequencies may be used instead; we refer the reader to [28,ch.Il] (our atomizing w i l l go on i n chapter I I I , when we are dealing with spectral synthesis). In contrast with both the r e a l and p-adic number f i e l d s (R has only the discrete subgroups hZ, where h £ R, and has no discrete subgroups at a l l ) , discrete subgroups abound in k{x and i t t u r n s out that: the introduction of harmonious sets here produces nothing new. We now confine our attention to V = k{x where harmonious sets may 24 be defined as follows. DEFINITION 1. A subset A of k{x ^ } i s harmonious i f for each (not necessarily continuous) character X o n k{x there i s a contin-uous character y with the property that x(^) = Y(A) for a l l X £ A. Characterizing these sets A in k{x ^ } i s very easy. Let T and G both denote k{x ^ } (so = G), and for each integer r l e t F r and G r both denote the subgroup {z £ k{x L(z) < -r} (elements of -r-1 -r-2 this subgroup have the form a^x + a^x + with the a's, of course, belonging to our p-element f i e l d k ) . By a discrete subgroup D of T , we mean a subgroup D such that D fl T r = {0} for some integer r We r e c a l l that the annihilator A"*" of a subset A of a group, i s the set of a l l elements i n the dual representing characters which are 1 on LEMMA 1. Let D be a subgroup of T, and l e t D"^" be i t s annihilator i n G . Then D fl ^  = {0} i f and only i f D 1 + G_r = G Proof. We can assume, for the proof, that D i s closed. Then D and D"*" are the annihilators of each other, as are r_ and G r _ r for each integer r . Now D O T = ( D 1 ) 1 0 G 1 = (D 1 + G ) 1 , r - r - r and Lemma 1 i s clear, since H = D"*" + G_r i s a closed subgroup of G , en t a i l i n g that H 1 = {0} i f and only i f H = G. 25 PROPOSITION 3. Let A c T. Then A i s harmonious i f and only i f A i s contained i n a discrete subgroup D of F . Proof. By [28, p.A3, T h . l ] , we know that A i s harmonious i f and only i f A"^" i s r e l a t i v e l y dense i n G (which means here that A"'" + G^ = G for some integer s ) . We l e t D = A"*" "*\ Then A c D and = A"K Now apply Lemma 1. A more general notion than 'harmonious set' i s that of 'coherent set of frequencies', which i s one of the central themes running through [28]. If P(z) i s a trigonometric polynomial an G, say P(z) = I a , Y (z)» z £ G, u€ F where F i s a f i n i t e subset of T, and the a u are nonzero complex numbers, then the elements of F are termed the frequencies of P. DEFINITION 2. A subset A of T i s a coherent set of frequencies i f there exist a constant C > 0 and a compact subset K of G such that sup | P | < C sup IP I G K for each trigonometric polynomial P on G whose frequencies belong to A The reader i s referred to [28, ch. IV] for the role played by such sets i n the study of almost periodic functions on a I.e.a. group. We have introduced them here for the characterization of the set T , i n terms of them, which we give in the next section. 26 §5. The connection with members of T Also i n here (§5.2): On the f r a c t i o n a l parts (A0 n) when 6 belongs to T page 30. r , G, r , ... remain as i n §4 (see the paragraph j u s t preceding Lemma 1 on page 24). From now on we s h a l l denote by , for each non-negative integer r , the f i n i t e vector space (over k) consisting of a l l polynomials i n k[x] having degree j< r : (22) Wr = {(D £ k[x]: L(co) < r}. In terms of the l i s t i n g tOg, ti)^, ... of k[x] given at (17) i n §2.2 r+1 of this chapter, we can write = {ion*. 0 <^  n < p }. §5.1 Characterizations of T. Theorem 1 below should be compared with-Prop. 7 and Th. IV on pages 58 and 110 respectively of [28]. THEOREM 1. Let 6 € k{x - 1} s a t i s f y 161 > 1, and l e t I be any f i n i t e subset of k[x] which includes 0 together with a basis for 2L(0 ) — 1 W2L(8)-1 ( t n i s h o l d s » for example, i f I z> {0,1,x, x } ) . Let A = A(6,I) be the set of a l l f i n i t e sums X.^ P.0 1 , where p. £ I i>0 i I for every i . Then each of the following statements implies the other two, (a) 0 belongs to T; (b) A i s harmonious ; (c) A i s a coherent set of frequencies . 27 Before proving Theorem 1, we make some remarks. 1°. The implication (b) => (c) holds i n general, and there i s nothing to prove here (see [28, p.110]). 2°. If 181 < 1 , then the set A = A(9,I) i s an i n f i n i t e bounded subset of T, and so by [28, p.114, Prop.3], cannot be a coherent set of frequencies. 3°. Clearly each subset of an harmonious set i s again harmonious, and Theorem 1 implies that i f 9 £ T and I i s any f i n i t e subset of k[x], then A = A(9,I) i s harmonious. For arbitrary 9 s a t i s f y i n g 191 > 1, we have that A = A(9,I) i s harmonious whenever I cr WT , Q X , L{v)-L ( i f I = W L ( 0 ) _ I > then A = A(8,I) i s a discrete subgroup of F whose intersection with the group ^ = (u e T: L(u) < 0} i s {0}) . 4°. Let A = A(8,I) be as i n Theorem 1. It i s not hard to see (cf. §1.5.2, to follow) that any u £ A"*" has the property that (23) I(u8 n)l < 181"2 for a l l n > 0 . Thus i f A"1" * {0} , then 8 belongs to T by Grandet-Hugot's theorem (in §1.3, page 22) and the implication (b) => (a) i s clear. The fact that (a) =» (b) means that A i s harmonious i f and only i f A"1" * {0}, and i f (23) holds for a single u * 0, and only for s u f f i c i e n t l y large n , then i t holds as written for the r e l a t i v e l y dense set of u belong-ing to A"*" . We w i l l expand a l i t t l e on this i n the next section. Note that i f 8 t T , then the subgroup of V generated by A(8,I) must be dense i n V . 28 Proof of Theorem 1. (a) => (b) . Suppose 9 £ T , and l e t s be the degree of 9 over k(x). If s = 1 , then 0 £ k[x] and hence A = A(9,I) i s contained i n k[x] , which i s harmonious. Assume s > 1 , l e t 0=0^, 9^, © s be the conjugates of 9 , and l e t 0 ^ = i d e n t i t y , , . . ., 0"^ be the corresponding d i s t i n c t embeddings (over k(x)) of k(x)(0) into the algebraic closure K of k{x }^ (thus each 0 fixes k(x) , and o\('e) = 6 , for j = 2, s ) . There i s an integer r > 0 such that A = A(0,I) i s contained i n the group A consisting of a l l f i n i t e sums Z. _ p.0 1, where p. £ W r i>0 I I r for every i (see (22)). We need only show that A^ i s discrete. Suppose A £ A , say A = Z™ . p.0 1. r J i=0 I Then A belongs to k(x)(0) , and A i s i n t e g r a l over k[x] , hence we may apply the norm function N = n_._^  to A , obtaining an element N(A) £ k[x] (see, for example, [45 , p. 260, Th. 4]) . Suppose A * 0 . Then N(A) * 0 , and hence |N(A)I > 1 . Now N(A) = A n ^ = 2 0 j ( A ) , and a (A) = Z ™ = Q p.,0..1 , where I0_. I _< 1 for 2 < j < s , since 0 belongs to T . Thus we have la. (A) I < max Ip.l < p r , 2 £ J _£ s > 3 i>0 1 from which we obtain (24) 1 < |N(A)I < IAI p r ( s _ 1 ) . We deduce that L(A) > - r ( s - l ) for every nonzero A in A^ , hence A r i s discrete and so A i s harmonious, by Proposition 3 in §1.4 . 29 (b) => (c) . See Remark 1° above. (c) => (a). Suppose that 9 £ T , l e t B e l be the basis for ^2L(Q)-l which we assume I contains, and define <Ku) = (2L(9)+1)" 1( 1 + I Y n(u) ) , u £ G. peB p There i s a constant T < 1 such that (25) <J>(u) = 1 i f l(u)l < 191"2 , Icf>(u)l < T otherwise. — 1 -2 To see (25), we use the fact that w 2 L ( e ) _ i a n d £k{x }: I(u)I < 181 } _2 are the annihilators of each other. Thus i f I(u)I < 181 , then c e r t a i n l y cj>(u) = 1. On the other hand, i f <J)(u) = 1 , then Y p(u) = 1 for every p £ B , and hence u belongs to the annihilator of WOT,as . Zh (<o) — i Evidently, the range of <J) consists of 1 together with f i n i t e l y many values s t r i c t l y inside the unit disk. Now define the sequence t})^ , k>l, of trigonometric polynomials on G by ,k * k(u) = nj = 1<))(u9 1) , u £ G. Evidently, the frequencies of each <J> belong to A . Now Grandet-Hugot's K. theorem (§1.3, page 22) and (25) show that l<t>k(u) I decreases to 0 as k -»- oo for every u * 0 ; hence ^ ( u ) "*" 0 uniformly on compact sets d i s j o i n t from 0, by Dini's theorem. Assume that there exist a compact subset K of G and a constant C > 0 such that sup 1 \JJi •< Csup|\J>| for a l l trigonometric polynomials G K \> on G whose frequencies belong to A . Choose U Q £ G such that u^ + K does not contain 0, and l e t <|/k(u) = c|>k(u+u0) , k > 1. 30 then each i s a trigonometric polynomial on G whose frequencies belong to A , sup 1^1 = 4^(0) = 1 for each k , and sup 0 G K as k •> 0 0 . Thus A cannot be a coherent set of frequencies. §5.2 On the f r a c t i o n a l parts (X0 n) when 8 belongs to T. We know from §1.3 that i f 9 £ k{x - 1} and |0| > 1 , and i f I(X8 ) I < 181 holds for a single X =t= 0 and a l l suff i c i e n t l y large n , then 8 belongs to T. Now suppose that 8 £ T , and that the degree of 0 over k(x) i s s ^  1 . Let N >_ 0 be an integer, and le t D be the subgroup of T = k{x ^ } consisting of a l l f i n i t e sums ^i>0 P i ^ 1 ' where £ WN for each i (see (22)). The analysis of the previous section leading up to (24) shows that D n r N ( s - l ) = { 0 } ' and Lemma 1 i n §1.4 (page 24) shows therefore that (26) G ^ ^ = G (this notation was fixed just prior to Lemma 1). 31 Suppose A £ D Then Y^(p6 n) = 1 for a l l n > 0 and p £ WN . It follows that A 8 n belongs to for every n > 0 , or put another way, (27) I ( A 9 n ) I < p- ( N + 1>, n > 0 . Now (26) shows that (27) i s s a t i s f i e d for some nonzero A s a t i s f y i n g |A| < p ^ s ^ , and we have proved PROPOSITION 4. Let 0 belong to T, and l e t s > 1 be the degree of 9 over k(x). Then for each integer W >_ 0 , there exists A £ k{x }^ , A * 0 , such that |A| < p N ^ s ^ , and such that I ( A 0 n ) I < p " ( N + 1 ) for a l l n > 0 . Notes. Proposition 4 w i l l play a role i n the next chapter when we deal with the uniqueness problem, but Theorem 1 i n §5.1 w i l l not be needed u n t i l chapter III when we are dealing with spectral synthesis. For comparison, l e t us mention that on the l i n e , t h e set of a l l powers {9^: k=0,l,2,...} of a r e a l number 9 > 1 i s a harmonious set of re a l frequencies i f and only i f 8 i s a Pisot or Salem number, but the set of a l l f i n i t e sums Z.^„ e . 8 1 , where £. = 0 or 1 for every i , 1>0 1 1 J forms a harmonious set i f and only i f 8 i s a Pisot number. The characterization of harmonious s e t s - i n k{x }^ which we gave i n §1.4 should be compared with the characterizations given in [28] for the r e a l and p-adic cases; these are somewhat more d i f f i c u l t to obtain! 32 §6. Character series on the compact group - and some probability  considerations. Throughout this section, and indeed right up u n t i l chapter III of this thesis, we s h a l l denote the compact group G simply by G: G = {z £ k{x L}: L(z) < 0} A character series on G i s simply a series (28) S = I . a y n=0 n 1n where the~ a are complex numbers, n Now (28) could be the (G-) Fourier series S[f] of a function f € L^(G) ; this means that the a^ are given by a n = f (n) , where f(n) = / f(z)Y (z)dz , f € L X(G) , G n and a necessary condition for this would be that a^ ->- 0 , by the Riemann-Lebesgue lemma. In chapter III we s h a l l be very interested i n such f where we have Z If(n) I < °° , but u n t i l then we are interested i n arbitrary series of the form (28), and especially their pointwise convergence. The N-th p a r t i a l sum of (28) i s the polynomial (29) S N = ^ : J a n Y n f N= 1, 2, .... And now for ju s t a l i t t l e p r o b a b i l i t y . 33 We r e f e r the reader to [10] for the elementary theory of independent random v a r i a b l e s , martingales, and so on. N.B. A l l a - f i e l d s discussed below w i l l be assumed to have adjoined to them a l l sets of Haar measure zero. Let HQ = {0} , and f o r each N _> 1 l e t IL^ denote the f i n i t e subgroup of G given by (30) IL^ = { z j a l a . x - 1 : ^ . . . . a ^ € k} , where k i s our p-element f i e l d . Now for each N > 0 , we have a pair of a - f i e l d s F„ and F' — N N defined as follows (remember that G^ denotes the subgroup {L(z) < -r} fo r each r > 0 ) : N (31) N F - the atomic O-field generated by the p cosets i n G of G„ N -N r ' - the p - p e r i o d i c a - f i e l d H + 8 , where N 8 i s the Borel f i e l d of G N The f i e l d F^  consists of p atoms, plus a l l n u l l sets, while F^  consists of a l l sets + A , where A i s a Borel subset of G -N (the p - p e r i o d i c s e t s ) . Note that F^  i s e s s e n t i a l l y j u s t the 1-atom f i e l d {d>,G} , while F^  i s the Borel f i e l d 8 i t s e l f . Put another way, FN i s the smallest complete a - f i e l d with respect to 34 which each of the characters y., Y » •••> Y N - l i s measurable (N > 1), i p p" A — while F^  i s the corresponding a - f i e l d for the characters YpN> YpN+1' •••• I f F i s a Borel s u b f i e l d of 8 , we s h a l l write f £ F to mean that f i s an F-measurable function. Now for each N >_ 0 , the f i e l d s F^  and F^  are independent, which means that we have (32) J f gdm = / f dm J gdm G G G whenever f and g are integrable functions with f £ F„ and g £ F' N ° N ( m i s normalized Haar measure). Suppose F i s a Borel s u b f i e l d of 8 , and f i s an integrable function. Then there i s an (essentially) unique function g £ F , denoted E(f I F) , with the property that / g dm = J fdm for a l l sets A A A belonging to F . Now the f i e l d s F^  , N >^ 0 , defined above are increasing: ^0 C ^1 C " " ' ^ s e c l u e n c e fg» ••• °f integrable functions with the property that f £ FN for each N , i s ca l l e d a martingale (with respect to, or adapted to, our sequence {FN} ; this w i l l sometimes be suppressed below) i f = E ( f N + ^ | F^ ) for a l l N ^  0 . What does a l l this have to do with us? Let f £ Ll(G) , l e t S [ f ] be i t s Fourier s e r i e s , and l e t S [f] n denote the n th p a r t i a l sum of S [ f ] for each n > 1 ( S Q = 0 always). Then (33) S p N [ f ] = E(f I FN) , N > 0 , and the uniformly integrable sequence S^N[f] , N>0 i s a martingale 35 which converges to f a.e. and i n (see, for example, [ 1 0 , p. 336, Th.9.4.5]). Any martingale bounded i n converges a.e. (this i s the c l a s s i c a l martingale convergence theorem of Doob), and any uniformly integrable martingale converges a.e. and also i n L^ " . CO Now return to the case (28) of an arbi t r a r y series S = Z _ a Y . J n = 0 n'n N and consider a p a r t i a l sum S where n > p . We have n <-> r-n—1 _ , T~n— 1 S = I. . a.y. = S N + I. w a . y . . n j = 0 J ' J P N J = P N ] ] If A i s an atom of (a coset of G^), then for each j i n the range 0 <^  j < p^ , 7 ^ i s constant on A , while for each j >^  p^, the i n t e g r a l over A of Yj i s 0 . In p a r t i c u l a r , one sees that (34) / S dm = / S N dm, n > p N , A £ F X T , \ n A P ~ N and the sequence S^ N , N _> 0 , forms a martingale with respect to the a - f i e l d s F„, N > 0 . N — Now Chow's martingale convergence theorem can be applied here ([ 9 ], see also [17, p. 242]): l e t N^ < N 2 < ... be a sequence of p o s i t i v e integers; then lim S M. exists and i s f i n i t e a.e. on the union of the sets P " J where lim i n f . S M . > - 0 0 and lim sup. S N . < 0 0 . J P 1 N J F J P ^ J N The p -th p a r t i a l sums of character series play a distinguished role i n harmonic analysis on G , and there i s surely more than a hint of this here! 36 Notes. What has just been presented i s of course very well known, and we have simply taken the opportunity to set some notation and a con-venient reference for some of our work i n chapter I I . Gundy [17] appears to have been among the f i r s t to exploit the prevalence of martingales i n certain systems which include ours. This prevalence was no doubt i n s p i r a -t i o n a l i n the subsequent discovery [18] of a (backwards) martingale i n the trigonometric system ( i . e . , on the c i r c l e ) . Character series on G are also studied from the point of view of 'quasi-measures' on G (also c a l l e d dyadic measures i n the case of p = 2) . See Yoneda [44] and Wade & Yoneda [41]. In t h i s , the martingale property expressed by (34) i s exploited. Let I be the f i e l d which i s the union of a l l of the f i e l d s F . A f i n i t e l y additive set function y on I i s c a l l e d a quasi-measure. Any quasi-measure y has a 'Fourier s e r i e s ' S[y] defined by CO ^ zv f S[y] = ZQ y(n)y n , where y(n) = J G Y n dy (this 'integral' i s j u s t the f i n i t e sum y (h)y(h+G w), where N i s h e % n N any integer for which n < p , and i s as i n (30)). oo Now every series S = I_ a Y i s the Fourier series of some quasi-0 n n measure, for given S, a set function y i s well-defined on I by setting y(I) = J S H for each I £ F„. j. p N (34) shows that y i s a well-defined quasi-measure, and i t i s easy to v e r i f y that S = S[y]. Quasi-measures whose transforms y are bounded are c a l l e d pseudo- measures , and i f y(n) 0 as n -> °°, y i s c a l l e d a pseudofunction. These two subclasses w i l l make f l e e t i n g appearences i n the next chapter, with a more substantial role reserved for the pseudomeasures i n chapter I I I . **** 37 §7. Formal m u l t i p l i c a t i o n of series - and the analogue of Rajchman's  theorem on the c i r c l e . What i s the behaviour of a character series S on the compact group G = {z: L(z) < 0 } i f S i s the formal product of two given series S and T on G ? Knowing the answer to t h i s , i n the case that T i s some trigonometric polynomial P , i s important i n the theories of uniqueness and l o c a l i z a t i o n on G since the c h a r a c t e r i s t i c function of a basic open subset (a coset of some G^= {z: L(z) < -r}, r >_' 0, i n G) i s j u s t such a P . The results presented i n §7.3 below are not new, but we give the pretty Proposition 5 a short proof and discuss some of i t s immediate consequences. The main new result of this section i s Theorem 2, which we give i n §7.4 ; this i s the (exact) analogue of Rajchman's theorem on products of trigonometric series on the c i r c l e . §7.1 Notation and preliminaries. In this section we follow the usual custom of indexing characters y on G with nonnegative integers . Let the symbols © and 9 denote p-series addition and subtraction of natural numbers respectively. In terms of our standard l i s t i n g to^, co^, ... of the members of k[x] ( e.( w n) = n , where e^  i s the mapping (3) ) , these operations are given by n © m = e(to + to ) , n 9 m = e(to - w ) , so that one also — n m — n m has co = to + to and to = to - to for integers n, m > 0 . n©m n m nGm n m 6 ' — Then we have too y Y = Y . and y y = y for n, m > 0. n m n©m n m n9m — It i s clear that one always has (35) k © j < k + j , k, j > 0 . 38 Now l e t us make some observations on the operation 9 . LEMMA 2. Assume k, j , N, r and s are nonnegative integers. Then (36) k 9 j < k + ( p - l ) j ; (37) j < s p N < r p N < k implies k 9 j > (r-s+l) p N ; (38) j < r p N < k or k < r p N < j implies k 9 j > p N ; (39) k £ A r = {j: r p N < j < (r+l)p N} implies { k 9 j : j £ Ar> = A Q . (Note that i n the Walsh case of p = 2 , the operations © and 9 coincide, and (36) reduces to (35) . ) To prove the lemma, suppose k and j have the unique (and f i n i t e ) expansions oo j[ 0 0 i k = Z 0 u i p ' ^ = Z 0 v i P ' 0 - U i < p ' 0 - v i < p ' and l e t us c a l l the u's and the v's the coordinates of k and j respectively. Now k © i has coordinates t. = u. + v. (mod p) , and (35) on the previous page i s clea r . Also, 9j (the p-series negative of j ) has coordinates v * given by v. l r p-v,- i f v. * 0 1 0 i f v. = 0 so that v* <_ ( p - l ) v i for each i > 0 , which y i e l d s Bj < ( p - l ) j . This combined with (35) proves (36) . To prove (37) , we write k = up N + h , j = vp N + i , 39 where the integers u and v s a t i s f y u >_ r , 0 ^ v < s , and where N N 0 < h < p , 0 < i < p . N N Now up and vp have no nonzero coordinates of index < N , while h and i have none of index _> N , so we can write k Gj = (up N 9 vp N) + (h 9 i ) = (u 9 v) p N + (h 9 i ) . For a l l nonnegative integers a and b we have, of course, a 9 b >^  0 , and also a 9 b _> a - b , since (35) enables us to write a = a 9 b © b < ( a G b ) + b . Thus we have k 9 j > (u - v ) p N > (r - s + l ) p N , and (37) i s proved. N Now (38) follows from (37) upon n o t i c i n g that 9 a > p whenever N the integer a s a t i s f i e s a ^ p . To see (39) , notice that the numbers i n {k 9 j : j £ A^} are N M a l l d i s t i n c t and less than p . ( A l t e r n a t i v e l y , A Q = {j : 0 < j < p } i s a subgroup under © of the group ( Z + , © ) , and A ^ i s the coset N rp © A Q of A Q . Thus i f k £ A ^ , then very c l e a r l y one has k 9 A = A _ . ) r 0 §7.2 Formal products of s e r i e s . In what follows, i t may help to think of the s e r i e s S, T, and S* as follows: 40 S = 0 n n (40) T = L b y 0 n'n c Y 0 n'n an arbitrary and possibly bad series, whose pointwise convergence properties on G are under investigation. S i s not necessarily a Fourier series, and may not even s a t i s f y a 0. a very good,absolutely convergent series, whose sum at z€ G w i l l be denoted by T(z) the series, also denoted ST, which i s ob-tained by formally multiplying S and T together. Now T(z) i s the sum of the series T. When we refer to the 'series' T(z)S , we s h a l l mean the series on G whose nth p a r t i a l sum at zG G CO i s T(z)S (z) . That i s to say, T(z)S i s the series E n T(z)a y (z), where n U n n CO £ S = Z A a y (z). The series S 'ought' to behave l i k e T(z)S. Two 0 n n ° series U and V on G are c a l l e d equiconvergent at some point i f the n th p a r t i a l sum difference - converges to 0 at that point. On any set where this convergence i s uniform, they are called uniformly  equiconvergent on that set. Let us mention now that Theorem 2 i n §7.4 below w i l l show that i f we have a n -»• 0 and I n lb I < 0 0 , then the series S i s uniformly equiconvergent with the series T(z)S . Before we come to that, however, l e t us investigate when the series S i s well-defined, and also deal with the r e l a t i v e l y simple polynomial case (the case where T i s a polynomial on G). 41 The c o e f f i c i e n t s c of S i n (40) are given by n ( 4 1 ) °n " * aiV iffij=n LEMMA 3. Let the series T i n (40) be absolutely convergent : Z |b I < °° . Then n (i) i f {a } i s bounded, S* i s well-defined and the c i n (41) n n are bounded; ( i i ) i f i n fact a ->• 0 , then c 0 also. n n Proof: with c = Z. a.b = I. n a ^ .b. , (i) i s obvious n j=0 j nOj j=0 nGj j and ( i i ) i s an easy consequence of observations i n Lemma 2. Pl a i n l y S i s always well-defined i f T i s a polynomial on G (so that (41) i s a f i n i t e sum). §7.3 Products with a polynomial: S = SP. PROPOSITION 5. Let S = Z n a Y be a series with arbitrary n=0 n'n 3 p N - l c o e f f i c i e n t s {a } , and l e t P be a polynomial on G, say P = Z _ b y where N ^  0 . Let S* be the formal product of S with P . Then the p a r t i a l sums (with a 0th p a r t i a l sum always 0) obey (42) S* N ( z ) = P(z)S N(Z) , z £ G, m = 0, 1,2, ... . mp1N mp ' •k oo oo Proof: We have S = Z „ C Y > c = Z . „ a . b _ . n=0 n n n j=0 j nGj 42 If r > 0 i s an integer, then ( r + l ) p N - l S* n N " S* N = I ( I a. b _. . ) Y (r+l)p N rp n = r p N j = 0 J nGj Tn ( r + l ) p N - l I a.Y. X b Q.Y Q . j-0 2 1 n=rp N n G j n 9 j ( r + l ) p N - l P N-1 I a.y. I b.Y. j=rpN 3 1 i=0 1 1 P . ( S , , n N - S N ) • (r+l)p rp Now sum both sides over 0 _< r < m to get (42) . Remark: no sums i n the proof were r e a l l y i n f i n i t e , and t a c i t use was made of Lemma 2 (page 38)• COROLLARY 1. Let S and P be as i n Proposition 5, but suppose i n addition that a -»• 0 . Then S i s uniformly equiconvergent on G with the series P(z)S : lim [ S*(z) - P(z)S (z)] = 0 uniformly for z £ G. n-x» Proof: immediate from Lemma 3 ( i i ) and Proposition 5. COROLLARY 2. Let S be a character series on G with arb i t r a r y c o e f f i c i e n t s , and l e t I be a coset i n G of G^ = {z: L(z) < -N} for some N :> 0. Then there i s a series S on G which s a t i s f i e s . r S N ( z ) , z £ I 43 Moreover, the c o e f f i c i e n t s of S remain bounded (or tend to 0) whenever those of S do the same. Proof: choose P i n Proposition 5 to be the c h a r a c t e r i s t i c function of I . Lemma 3 t e l l s about the c o e f f i c i e n t s . COROLLARY 3. Let S be a character series on G with c o e f f i c i e n t s tending to 0, and l e t I be a basic open subset of G (a coset of some GN ' N j> 0 ) . Then there i s a series S on G which i s uniformly equiconvergent with S on I , and which converges uniformly to 0 on the complement of I . Proof: clear from Corollary 2. To give one more i l l u s t r a t i o n of the power of Proposition 5, we w i l l use the early uniqueness re s u l t of Vilenkin [37] that a series which converges everywhere on G to 0 must be the zero series ( i . e . , a l l i t s c o e f f i c i e n t s are 0 ) . COROLLARY 4. Let a series S converge to 0 on a coset I of G.T , where N > 0. Then a l l of the p a r t i a l sums S N , m = 0, 1,2, ... N — r mpN ' ' ' vanish on I (so i n p a r t i c u l a r , S -»• 0 uniformly on I ). n L-Proof: With P = 1^ . i n Proposition 5, the series S converges everywhere on G to 0, hence i t i s the zero s e r i e s . Now (42) gives the r e s u l t . Corollary 4 w i l l have a role to play i n §11.1.2 when we deal 44 b r i e f l y with the relationship between pseudofunctions and the uniqueness problem. §7.4 Ra.jchman's theorem for products ST. We now present a r e s u l t which i s stronger than Corollary 1 of the previous section, and which looks more l i k e Rajchman's theorem for the case of the c i r c l e . The r e s u l t i s new, even for p= 2. To get i t , we must break up our sums i n the right way, and use a combinatorial argument at one point. oo THEOREM 2. Let the series S = Z n a y s a t i s f y a -> 0, and the 0 n n J n oo ft series T = I 0 b Y s a t i s f y I n | b I < 0 0 . Let S „denote the formal 0 n'n J n product of S with T , and denote the sum of the series T at z £ G by T(z). Then (43) l i m [ S * ( z ) - T(z)S (z) ] = 0 uniformly on G. n n n-*» Proof. Assume rp 1^ < n < (r+1) p N , where 1 _< r < p . 45 Fix zn £ G , and put a. = a.Y.(z_.) for i > 0 . Let 6 Q = b Q - T(z Q). , and 6.. = b..Yj(z 0) for j > 1 . Then co n _ i s ; < z o > - T ( * o > W " z z aAoi J=0 K=0 r p N - l n-1 oo n-1 1 1 + Z Z j=0 k=0 j=(r+l)p N k=0 ( r + l ) p N - l r p N - l ( r + l ) p N - l n-1 + I I + I I I a . B , „ . N i n N , N / j k9j j=rp 1 N k=0 J=rp k=rp ' J J = + a 2 + + a4 ' say. In the estimates below, presents the most d i f f i c u l t y . Let A = SUP^Q I CIJ I = sup g I I , and notice that (44) Estimates follow. r p N - l (a) ax = - a. ^ B k g j ^ u s i n g ( 4 4 ) _ R e r e > k Q . > p N (38) (displays (36)-(39) are i n Lemma 2 on page 38), and so l a , I < A r p N Z |b.I. n-1 oo (b) a„ = Z Z „ a - B i , Q - > a n d again by (38) we have k=0 j = ( r + l ) p N J k H j k9j > p , so l a . l < A(r+l)p N Z |b. I 2 ~ N i i>p 46 (c) S i m i l a r l y , 0"^  r eadily admits the estimate 1-0,1 < A p N Z lb. I. 3 — _ N l i>p ( r + l ) p N - l n-1 N , N J k0j j=rp i N k=rp J J p -1 r n-1 Z X a. . . n i N k 9 i i=0 Lk=rp B. i upon a change of variables and use of (39). We estimate this e s s e n t i a l l y using n-1 n-1 Z OL _ . = Z a. + an error E. . N k 9 i , N K i k=rp" k=rp" Put E± « {k9i : r p N < k < n-1} for 0 < i < p N. Then (36) and (37) imply that (45) E ± c {rp", n-l+(p-l)i}, Let Then a. (i) f +1 i f j € E ± ^ E 0 -1 i f j E E 0 " E i PN-1 Z a. i=0 j£E PN-1 r S, + Z Z e a ) a . i=0 L jeE-jAEQ J Now y, = - Z ex.. Z i t B, by (44), so 0 N ly, I < A P " z l b , I l >p Notice that (45) implies |E^ A E Q I < 2 ( p - l ) i , so ly,I < 2(p-l).max l a . I . Z i l b . I 2 " j>P N 1 i - 1 1 47 Our estimates (a), (b), (c) and (d) add up to no more than oo 4Ap Z „ i I b . I + 2p max I a . | . I i I b . I i>p J>P J i=l which tends to 0 as N -*• 0 0 by the hypotheses of the theorem. This completes the proof of Theorem 2. Notice that our f i n a l estimate s t i l l gives us a r e s u l t i f we assume merely that the a^ are bounded. Notes. The sum i n this proof was running away from us, but f i n a l l y we caught i t with • W e a r e a o t . however, aware of any use at a l l for Theorem 2 at present where the results of 57.3 are not adequate for the purpose. Possibly there are connections with the idea of a derivative on G; our series T can be viewed as one having an absolutely convergent 'derived' s e r i e s . We w i l l paraphrase Theorem 2 s l i g h t l y i n §11.1.3(page 56). A discussion of the 'dyadic d e r i v a t i v e ' w i l l be found i n [401 . Notice that the condition on the series T i n Theorem 2 i s equivalent to N+l , oo P -1 / / 1 N 'V < " ' N=l n=p and that the condition i s certainly f u l f i l l e d i f T i s a polynomial on G as i n Corollary 1 of §7.3. The results of §7.3 are due largely to Sneider [35] i n the case of p = 2 , and to Wade [39] i n the more general case. Proposition 5 i s comparable with Rajchman's theorem on the c i r c l e i n i t s applications to uniqueness and l o c a l i z a t i o n problems. The more c l a s s i c a l case of the c i r c l e can be found dealt with i n [46, vol.1, ch.IX]. 48 CHAPTER II UNIQUENESS OF REPRESENTATION Given a character series S observed to 'represent' an integ-rable function f i n some way, when can we conclude that S must be the Fourier series of f ? I t i s a remarkable fact that a series may converge almost everywhere to zero, yet not be the zero series (the Fourier series of 0). Several examples of such series w i l l be found as we explore the problem i n this chapter. In §11.6 we w i l l see just how subtle the answer can be. §1. The problem of uniqueness. also contained i n this introductory section: pseudofunctions and uniqueness ( § 1 . 2 ) . . . . p.51 a s u f f i c i e n t condition for a set E to be a set of uniqueness (§1.3) P-57 Throughout this section (and indeed the entire chapter) G w i l l denote the compact group {z e k{x ^}: L(z) < 0}, and E can generally be thought of as some 'thin' subset of G which w i l l usually, but not always, be closed. §1.1 Sets of uniqueness and sets of m u l t i p l i c i t y . DEFINITION 3. A subset E of G i s c a l l e d a U-set (set of unique-00 ness) i f the convergence to 0 of a series S = I Q a n ^ n o u t s i d e E implies a =0. A set E which i s not a U-set i s call e d an M-set (set of mult-n i p l i c i t y ) V i l e n kin [37] showed that a series S which converges everywhere 49 to 0 on G must be the zero series, and thus the empty set i s a U-set (this corresponds to the o r i g i n a l uniqueness r e s u l t of Cantor for t r i g o -nometric series E a e i n X on the c i r c l e ; see, for e.g., [46, vol.1, p.326]) . —OO ft Now i f E i s any set of positive Haar measure m(E) > 0, then E must contain a closed subset F with m(F) > 0 also, and the Fourier series of f = 1 i s n o n t r i v i a l . This series w i l l converge to 0 outside r F (§11.1.2 below) and hence also outside E . In fact the following two statements are well-known: (a) a l l sets of po s i t i v e Haar measure are M-sets; (b) any countable subset of G i s a U-set. We are l e f t with the uncountable sets of measure 0 to look at, and even among the perfect sets of this class, characterizing the U-sets i s a del i c a t e matter. Haar measure cannot discern a l l M-sets. Let us consider for a moment a n u l l series S * 0, by which we mean a nonzero series converging a.e. to 0. Can such an S be the Fourier series of some integrable function f ? The answer i s no, because as noted i n §1.6, S = S[f] implies S p^ ->- f a.e.. But S i s always the Fourier series of some- thing (cf. §1.6 Notes), and since i t s c o e f f i c i e n t s must tend to 0, S i s in fact the Fourier series of a 'pseudofunction' (§1.6 Notes, and §11.1.2 below). This gives a clue as to how we might proceed to find M-sets, apart from the ones possessing positive measure. Suppose E c: G and m(E) = 0, and suppose u belongs to the space M(E) of measures supported i n E. The Fourier series of u i s the series Sty] = I™ y(n)Y n , where y(n) = / Y ndy. 50 P l a i n l y a necessary condition for S[y] to converge to 0 outside i s that u(n) •* 0 as n -*- 0 0. Is the condition s u f f i c i e n t ? Indeed i t i s , and i n fact a l l of the perfect n u l l M-sets E that we f i n d i n this chapter are obtained as M-sets because they carry measures whose trans-forms vanish at i n f i n i t y . DEFINITION 4. A subset E of G i s an M^-set (or M-set in the r e s t r i c t e d sense) i f there i s some measure y on G (#0) whose Fourier series S[y] converges to 0 outside E. A set which i s not an M^-set i s c a l l e d a U n-set ( U-set i n the wide sense). PROPOSITION 6. If E c G and y £ M(E), then Sty] converges A, to 0 off E i f and only i f y(n) -»• 0 as n °°, in which case E i s an Mg-set (provided y * 0) . This w i l l be proved i n the next section. Let us add v a l i d i t y at this point to the use of the term 'U-set' by mentioning that i f a series S converges everywhere outside of a closed U-set to some f i n i t e integrable function f, then necessarily one has S = S[f] (the reader i s referred to the survey i n [381). 51 §1.2 Pseudofunctions and uniqueness, We suppose in this subsection that E i s closed. Let y be a pseudomeasure on G; that i s , assume y belongs to the space PM of continuous l i n e a r functionals on the Banach algebra A(G) of functions with absolutely convergent Fourier series. A member of A(G) has the CO form f = Z_ a y , with Z |a I < 0 0 , and the norm i n A(G) i s given u n n n by || f U A ( G ) = 1 | a n> > t h a t i s f l l A ( G ) = Z j f C n ) ! . We write < f,y> for the value of y at f € A(G). Now the Fourier transform y of y i s given by (46) y(n) = <Yn>W>, n = 0,1,2, and the Fourier series S[y] of y i s the series (47) S[y] = Z Q y(n)y n . 52 One has immediately then the Parseval r e l a t i o n < f,y> = <f,y> = f(n)u(n), and the PM norm of y i s given by II y l l p M = sup |y(n) I . n Series of the form (47), where y i s a pseudomeasure, f i l l out the class of character series with bounded c o e f f i c i e n t s . Let us agree to c a l l a set I c G a basic open set i f I i s a coset of some = {z: L(z) < -N}, N >_ 0. The c h a r a c t e r i s t i c function p N - l 1_ of such an I belongs to A(G), and has the form 1 T = Z a y . I I 0 n n We can define y(I) = <l].,u> for a l l I belonging to the f i e l d of sets I generated by basic open sets. The support of y i s the smallest closed set E with the property that <f,y> = 0 whenever f £ A(G) vanishes on some neighbourhood of E ( i . e . , whenever f i s supported i n the complement of E ) . Clearly then, i f y i s carried by (has support contained in) E , one has y(I) = 0 for a l l basic open sets I d i s j o i n t from E. Conversely, i f y(I) = 0 for a l l I which are d i s j o i n t from a set E , one can show that y must be supported i n E , so this gives an alternative d e f i n i t i o n of the support of a pseudomeasure. We c a l l y £ PM a pseudofunction (and write y £ PF ) i f y(n) 0 as n -»• <>° . 53 There are natural ways of regarding functions ( in L (G)) as measures, measures as pseudomeasures, and with these, functions become pseudofunc-tions (the Riemann-Lebesgue lemma). We write PM(E) and PF(E) for the spaces of pseudomeasures and pseudofunctions which are supported i n E. The purpose of th i s s ection i s to give the following characterizations (often taken as the d e f i n i t i o n s i n a general I.e.a. group) of closed U-sets and Ug-sets i n G. PROPOSITION 7. Let E c G be closed. Then (a) E i s a U-set i f and only i f PF(E) = {0}; (b) E i s a U Q-set i f and only i f M(E) fl PF(E) = {0} . The 'only i f part of (b) gives us Proposition 6. Before proving Proposition 7, l e t us make some observations. For any y 6 PM and closed set E c G, we have that y i s c a r r i e d by E i f and only i f y vanishes on a l l basic open sets I which are d i s j o i n t from E ( i . e . , i f and only i f y annihilates a l l f € A(G) which are supported by such I ) . Now for any a £ A(G), we can define the pseudomeasure ay by s e t t i n g < f,ay > = < a f ,y > for functions f £ A(G). We see e a s i l y that y i s c a r r i e d by E i f and only i f l^y = 0 for a l l I which do not i n t e r s e c t E. 54 LEMMA 4. Let y £ PM have the Fourier series S = S[y] , and l e t E c G be closed. Then y i s carried by E i f and only i f for each N _> 0 , vanishes on cosets of G^ which are d i s j o i n t from E. We begin the proof of Lemma 4 by noting (see (4) on page 10) that i f I i s a coset of G„ , then for any z„ £ I we have N 0 -N- PN"1 -(48) lz(z) = P £ n = Q W Y n ( z ) » so -N PN"1 K1I'V> = P Zn=0 V z 0 ) < Y n ' W > -N _P N-1 . . . -N „ , . = p I y(n )Y„(z n) = p S N ( z n ) . n=0 n 0 P 0 This, which may also be written as (49) y(I) - p " N S p N ( z 0 ) , I = z Q + G N , shows that Spjg vanishes on I i f I i s d i s j o i n t from the support of y. Some remarks are i n order before we complete the proof. 1°. The remark made e a r l i e r about an alternative d e f i n i t i o n of the support of a pseudomeasure i s j u s t i f i e d by (49) and Lemma 4. 2°. If we know only that y vanishes on the coset I of G , 55 where N > 0, then Lemma 4 implies that S r vanishes on I for a l l r >^  N, and Corollary 4 i n §1.7.3 (page 43) shows then that i n fact S vt = 0 on I for a l l m = 1, 2, .... mpN 3°. Much weaker conditions, such as ' lim = 0 on the complement of E', w i l l ensure that y i n Lemma 4 i s c a r r i e d by E , however we s h a l l not elaborate here; Lemma 4 i s s u f f i c i e n t f or our purpose. Proof of Lemma 4 (continued). I t remains to show that the condition on S = S[y] i n the lemma implies that y i s c a r r i e d by E . Suppose some basic open set I = z^ + G^ i s d i s j o i n t from E . We conclude the proof by showing l^y = 0. Just as i n Remark 2° above (but we do not assume y vanishes on I ! ) , the condition on S guarantees (50) SmpN( z 0 ) = 0 » m = 0, 1, 2, . Now the transform of l^y i s the convolution A A A and, as noted e a r l i e r (see (48)), we have - N - / x n N f P Y „ ( z n ) , 0 < n < p I I 0, n > p N . N N For any m = 0, 1, 2, ... , and n such that mp <^  n < (m+l)p , we compute ('9' i s p-series subtraction of numbers) 56 A A (m+l)p -1 A A 1 * y (n) = p N Z N y ( j ) l ] . ( n 9 j ) j=mp = P _ N Y n ( z 0 ) t S ( m + 1 ) p N ( z 0 ) - S m p N ( z 0 ) ] = 0 , by (50) . Thus l^.y = 0 , and the proof of Lemma 4 i s complete. Proof of Proposition 7. Let S be an arbitr a r y series with bounded c o e f f i c i e n t s , and l e t y be the pseudomeasure for which S = S[y]. Thus y i s arbitrary too, and S * 0 i f and only i f y * 0. Suppose S converges to 0 on the complement of E. Then i t s coeff-i c i e n t s tend to 0 (assuming E * G), so that y i s a pseudofunction. Corollary 4 i n §1.7.3 (page 43) shows more than enough for us to conclude v i a Lemma 4 that y i s carried by E. Conversely, i f y £ PF(E), then as already noted i n Remark 2° above, Lemma 4 and Corollary 4 i n §1.7.3 imply that S = S[y] s a t i s f i e s S m pN(zg) = 0 for m = 1, 2,3, ... provided z^ + G^ doesn't intersect E. With y(n) ->- 0 , we are able to conclude that S (z„) -*• 0 as n -»• °° n U Summing up, we have that S converges to 0 on the complement of E i f and only i f y £ PF(E). Parts (a) and (b) of Proposition 7 are now both clear. Let us conclude this section with a paraphrasing of Theorem 2 i n §1.7.4 (Rajchman's theorem). If y i s a pseudomeasure and a belongs to A(G), then as we have just seen, we may form the product ay, which 57 i s again a pseudomeasure (and i f y i s a pseudofunction, then so i s a y ) . Now the Fourier series of ay i s given as a product: S[ay] = S[a].S[y]. Let us introduce some notation. Suppose we have a pair a £ A(G) and CO 00 ^ y £ PM, say a = ZQ b^Y^. and y ~ Z^ y ( i ) y ^ , and l e t us suppose not only that a belongs to A(G) , but that Z i l b J < 0 0, so that the function • CO • a = ZQ i b ^ y ^ belongs also to A(G) ( a i s some kind of derivative of a i f you like) . For each N ^  0, l e t OL^ and y^ be respectively the mem-N ber of A(G) and the pseudomeasure defined by the p -th p a r t i a l sums of the nN . DN . • • P — 1 P — 1 ^  Fourier series of a and y: Q L t = Z i b . Y . and y„ ~ Z y ( i ) Y . • N Q I I N Q 1 Here i s what we ended up actually proving i n §1.7.4 : sup I S [ay](z) - a(z)S [y](z) I N n n n>p z€G < A l l y l l ^ M a - ^ l l ^ + B l l a | l A ( G ) l l y - y N l l p M for constants A and B depending only on our fixed prime p. For a £ A(G) and y £ PM, the right hand side i s bounded. I f i n fact y(n) -»- 0, so that y £ PF, then the right hand side tends to 0 . §1.3 A s u f f i c i e n t condition for a set E to be a U-set. Although we have not yet given any examples of M-sets of measure 0, we have at least seen how such sets might be found. The following theorem plays a role i n the next section, where we w i l l sec examples of perfect sets of uniqueness. See Bary [2, vol.11, p.377] for the corresponding result on the c i r c l e . 58 THEOREM U. A s u f f i c i e n t condition that the subset E c G be a U-set i s the existence of a sequence of functions X.(z) = Z°°_n b ( j ) T (z) , z £ G, j = 1,2,... , j n—U n n each s a t i s f y i n g n I b ^ ^ I < 0 0 and vanishing for z £ E, whose coe f f i c i e n t s s a t i s f y three properties: ( i ) Zn=0 l b n j ) | ± C < °° ' j = l j 2 ' ••• 5 ( i i ) |b£ j )l > A > 0 , j = 1, 2, ... ; ( i i i ) lin> b ( j ) = 0, n = l , 2, .... 3  n Remarks. In the case that the X are a l l polynomials on G (i) ^ ( b =0 for s u f f i c i e n t l y large n ) , this r e s u l t i s due to Sneider [35] n for the Walsh case p=2, and to Wade [39] for the case of a l l primes p. The strengthening here i s made possible by Theorem 2 i n §1.7.4, and Theorem U looks now j u s t l i k e i t s counterpart on the c i r c l e . Its f u l l strength won't i n fact be needed here; t y p i c a l l y in applications the X are c h a r a c t e r i s t i c functions of basic open sets I d i s j o i n t from E, so the polynomial case i s s u f f i c i e n t . Let us mention that since each i s continuous on G , there i s no loss of generality i n Theorem U i f we assume E to be closed (the closure of a set E f u l f i l l i n g the conditions w i l l also f u l f i l l them). Also, we need suppose only that each X i n Theorem U vanishes for z £ E^U. , where U. i s some U-set; this w i l l be seen i n a moment. 3 3 If the condition Z n l b ^ ^ | < °° for each j i s relaxed to merely 59 demanding that each A_. belong to A(G) , Theorem U i s s t i l l v a l i d provided the A are are known to vanish on neighbourhoods of the closure of E J ( v e r i f i c a t i o n of this we leave to the reader). Proof of Theorem U. Let S = Z_ a Y be some series which converges 0 n n to 0 outside E. Then for each fixed k :> 0, the series which i s the product of S with the character y^ does exactly the same (by the results of §1.7). Now the constant term of the l a t t e r series i s simply a^ . Thus i t s u f f i c e s f or us to prove that the constant term a^ of any such given S must be 0. Let S. be the formal product of S with A. for some fixed i > 1. J J By Theorem 2 in §1.7.A, S* i s uniformly equiconvergent on G with the series A (z)S. This l a s t series converges everywhere on G to 0 ( lim A.(z)S (z) = 0 for z € G ) , hence S* must be the zero s e r i e s . n-x» 2 n j In p a r t i c u l a r , i t s constant term a * ( j ) I s 0• Thus 0 = a * ( j ) = I 0 0 n a b ( j ) 0 J n=0 n n = a.bj» 4- Z . a b ( j ) . 0 0 n=l n n Conditions ( i ) , ( i i ) , ( i i i ) i n Theorem U, together with the fact that a^ 0 as n °°, show ea s i l y now that a^ = 0. The proof i s complete. Notes. We are not aware of any study of how transformations on G affect U-sets and M-sets. If the transformation i s a translation z^-z^+z, then i t i s v i r t u a l l y t r i v i a l that U-sets translate to U-sets. What happens with a d i l a t i o n z ->- az on G ? A set E c G which i s taken outside of G by such a d i l a t i o n can always be sent back v i a the natural homomorphism of k{x 1} onto G . If E i s closed, ctE w i l l remain a U-set i f E 60 i s a U-set; this i s not too hard to see. On the c i r c l e [0,1), Zygmund and Marcinkiewicz (see [46, vol.1, p. 350]) showed that i f E i s any trigonometric U-set, and i f 9 > 0 i s such that 9E i s also contained i n [0,1) (e.g., 0 < 9 < 1), then 0E i s also a U-set. We think that the same is true i n the p-series setting, although we are unable to point the reader to any proofs in the l i t e r a t u r e for this case. In addition to [46], Bary [2] contains a comprehensive treatment of the uniqueness problem for the c l a s s i c a l case of the c i r c l e . See [14], for e.g., for the modern viewpoint. Wade [38] should be consulted for a brisk survey of the l i t e r a t u r e and results i n the p-series case. ooOoo §2. Classes of perfect U-sets - and discussion of a general problem. in which we f i n d contained: sets of the type H ^ ( § 2 . 2 ) p. 62 a class of perfect U-sets (§2.3) p. 64 discussion of a general problem (§2.4)-with talk of sets of the 'overlap' and 'non-overlap' type. . . . p. 68 61 §2.1 Example of a perfect U-set. For s i m p l i c i t y , suppose that our prime here i s p ^ 3 , and consider the following perfect subset of G = {z £ k{x *}: L(z) < 0}: (51) E = {I e x : e ± = 0 or 1} . E has measure 0 since P >. 3 ( i f p = 2, then E i s a l l of G) . Notice that for any z £ E and integer j , the f r a c t i o n a l part (x^z) of x^z again l i e s i n E. In p a r t i c u l a r , i f A denotes the coset of Gl = {z: L(z) < -1} A = 2 x - 1 + G L then we see that for z £ E, the sequence x z, j = 1,2, ... never belongs to A, mod 1 (mod 1 i n this thesis w i l l always mean we are con-sidering only f r a c t i o n a l parts). Now the c h a r a c t e r i s t i c function of A i s -1 P-l X(z) = p Z a Y (z), n=0 n n where a Q = 1 (and i n fact a^ = e^ 7 r i n/P } 0 < n < p) . From what we have observed, the functions i n the sequence X (z) = X ( x J z ) , j = 1, 2, a l l vanish for every z £ E, and X i s given by -1 P-l X.(z) = p Z a y i ( z ) . J n = 0 n 'npJ The conditions of Theorem U i n the previous section are eas i l y seen to be f u l f i l l e d by these X. with respect to E , and so we obtain 62 that E i s a U-set. To see an example for the case of p = 2, consider r-r-00 - 2 l „ , 1 E = {I. . e.x : e. = 0 or 1}. 1=1 x 1 Here, we have x ^ E = E (mod 1) for any j = 1, 2, Such sets are sets of the type H ^ §2.2 Sets of the type H ^ DEFINITION 5. A sequence of vectors V ( m ) = ( v j m \ ... , v ^ ) i n k [ x ] n i s ca l l e d normal i f for every nonzero vector (a^, a ) i n , r , n , , _n (m) , , k l x l , the inner product L. , a.v. ->• °° m absolute value as m 0 0. 3 = 1 3 3 A subset E c G i s said to be of type H ^ i f there i s some non-empty open set A <= G n and a normal sequence V^ m^, m > 1, in k [ x ] n such that the vector T7(m) , , , (m) (m) . V v (z) = (vj z, .. . , v^ z) never enters A, mod 1 for z € E and m = 1, 2, ... (mod 1 here means we are considering only the f r a c t i o n a l parts of the components y ^ z ). PROPOSITION 8. Let E c G. If E i s of type for some n > 1, then E i s a U-set. H ^ - s e t s may be simply ca l l e d H-sets; such sets were introduced on the c i r c l e by Rajchman, where they are defined as follows. A subset E of the c i r c l e [0,1) i s an H-set i f there i s a sequence of integers 63 < < ... and an open i n t e r v a l I c: (0,1) such that n_.x ? I, mod 1 for x £ E and j = 1, 2, ... (the Cantor ternary set constructed on the unit i n t e r v a l i s an example of an H-set on the c i r c l e , with respect to the sequence 3 n, n = 1, 2, . . . ) . Wade [39] carried the d e f i n i t i o n s over to the p-series case and proved Proposition 8. We sketch the d e t a i l s (our notation i s somewhat di f f e r e n t from that used i n [39]). The idea of the proof i s exactly the same as i n the example we have just given i n §11.2.1. Proof of Proposition 8. Let co^, co^, ... be our standard l i s t i n g of k[x] : .e(w ) = J» J > 0 ( s e e (3))« Suppose E c G i s of type • , , -. ,T(m) . , r -,n „(m) , (m) (m). wxth respect to the normal sequence V i n klxj , V = (v , ..., v ) 1 n We can fi n d an integer N > 1 and cosets I,, I of G„ i n G — 1 n N (G N = {z: L(z) < -N}) such that, f o r z £ E and m > 1, there i s at least one i £ {1,2, ...,n} for which (v^ m^z) £ The c h a r a c t e r i s t i c function of I. i s i 1 I ± = p " N ( i + z ^ " 1 4%), 1 - i , ....», where the a r e certain p-th roots of unity. For each m >_ 1, the function (52) A (z) = n 1 (v(m)z) i = l X i 1 vanishes on E . Expanding A^ as a trigonometric polynomial, we see that i t s constant -nN term i s p , and we note that for k _> 0, y (vz) = y (z), where K r r = _e(w^v) . We can i n fact write 64 I d < P N where (a) 0 = (k^, .. . , k^) has components of nonnegative integers, with 10" I = max k. , l l (b) a„ = S 4i} ( a ^ i } = 1) , and a 1=0 k i U (c) ^ ( a ) = ( j i « V i n ) ) The normality of the sequence i s just what i s needed to apply Theorem U of §11.1.3 to the A : d> (a) -*• °° as m ->- » for a l l 0 * 0 . m m §2.3 A class of perfect U-sets. THEOREM 3. Let 8 € k{x _ 1} s a t i s f y 181 > 1 , but be otherwise arbitrary. Let I be a proper subset of { p £ k[x]: I pi < 181}.- Then the perfect subset of G CO — - i (54) E(8,I) = { E. = 1 e ±e : e ± £ i} i s a set of uniqueness for G. Remark. If I = {p £ k[x]: I p l < 181}, then E(6,I) i n (54) i s a l l of G. If p = 2 and 101 = 2 ( i . e . , L(0) = 1), then I i n Theorem 3 must be {0} or { l } . i n a l l other cases, the sets (54) are uncountable and (53) 65 indeed perfect. COROLLARY 5. Let 8 £ k{x~ } s a t i s f y 181 > 1 (or i n the case p = 2, 181 > 2), but be otherwise a r b i t r a r y . Then the perfect subset of G (55) E(8) = {Z? . e.8 - 1: e. = 0 or 1} i=l I l i s a set of uniqueness for G. F i r s t l e t us prove Theorem 3 i n a s p e c i a l case which includes Coroll-ary 5. The proof here i s easy, and f o r e t e l l s a l i t t l e of what i s to come in §11.6. Presently we w i l l view Theorem 3 i n a broader context and see just why i t i s true. LEMMA 5. Let 6 £ k{x and 181 > 1. Then there exists X £ k{x *} with IXI = 1, such that (56) I(X8 n)I < 181 1 for n = 0, 1, 2, where (X8 n) denotes as usual the f r a c t i o n a l part of X8 n. Lemma 5 should be compared with what we were able to say i n Propo-s i t i o n 4 (§1.5.2) when 8 belongs to T. Proof of Lemma 5. Let D be the subgroup of k{x ^ } consisting of a l l f i n i t e sums I . , . p . f l 1 , where p. belongs to VL / n . , = i>_0 i I b L(8)-l {p £ k[x]: |p| < 181} for each i . One sees that the inter s e c t i o n 66 of D with the group G = {z: L(z) < 0} i s D O G . ? {0}, and taking perpendiculars as i n Lemma 1 of §1.5, we obtain (57) D 1 + G = k{x _ 1}. Now (57) shows there i s A £ k{x "*"} , IAI = 1 , such that A annihilates D : Y^(p0 n) = 1 for a l l p £ W L ( 0 ) _ I A N C * n = 0, I, 2, This means A 9 N £ W ^ Q ) . ! = {z £ k{x - 1}: I (z) I < | 9 | - 1 } for n > 0, and Lemma 5 i s proved. PROPOSITION 9. Let 9 and I be as i n Theorem 3 , but suppose i n fact that there i s some b £ {0, 1, p-l} such that no member of I begins b x ^ ^ ^ + . . . (members of I have generally the form a 1 x L ( 8 ^ " 1 + ... + a ^ ^ ) . Then E = E(0,I) i n (54) i s of type H*"1 oo —- i Proof of Proposition 9. Suppose z = Z. _ e . 9 belongs to E. c J=0 3 With A £ k(x }, | A | = 1 having b een chosen so that (56) holds, consider, for n > 0, n 1 1 - 1 1 0 0 - i (58) A 8 = X e . A 0 - 1 + Z e ^ . A 9 J . j-0 n " J j = l n + J Taking f r a c t i o n a l parts i n (58) , we note that (56) implies that (eXQ3) £ Gl = {z: L(z) < -1} for a l l j > 0 and e £ I , while I e X e - " ' I < I O l " 1 i s p l a i n l y true for a l l j > 2 and e £ I . 67 It follows that the coset in G of G^ to which ( X 0 nz) belongs i s completely determined by e n + i A Q ^ o t n e r words, the c o e f f i c i e n t of x ^  i n X 0 nz i s i d e n t i c a l with that of e ,.A6 ^ ) . n+l Our assumption on I i n Proposition 9 means that there i s some coset A = ax * + G^ of G^ to which eX0 ^ doesn't belong, for any e £ I, and thus (X0 z) I A f or any n > 0 . Clearly the same i s therefore true for , ( [ X 0 n]z) , and we have that E (0,I) i s of type with respect to the sequence of i n t e g r a l parts [ X 0 n ] , n = 1, 2,3, ... . Remark. We do not know i f a l l of-the sets E(8,I) i n Theorem 3 are of type but they are certainly a l l U-sets of a very elementary type, as we now show. Proof of Theorem 3. Fix 0 £ k{x , 101 > 1 , and suppose that I c W L ( e )_ 1 = { p £ k[x]: | p | < | 0 | } , with I * W L ( e )_ r oo - i L ( 0 ) Every z £ G can be written uniquely i n the form z = Z. , p.x i=l l where p_^  £ w L ( Q ) _ ^ f ° r every i . Let <f) be the mapping , °° -iL ( 0 ) °° 0 - i <$>: Z. p.x -> Z. p .0 . 1 l 1 l Then (j) i s a homeomorphism of G onto G , and a group isomorphism as well. Let E be the compact subset of G defined by / T - 0 0 - i L ( 0 ) t 1 E = {Z. . e.x : e. £ 1}. i=l l l Evidently, (f>(E) i s the set E (0,I) i n ( 5 4 ) . It i s clear that E i s a set of the type with respect to the sequence x ° L ^ , n = 1, 2, . . . and the open subset A of G given by 68 e l C = w L ( Q ) _ 1 N " 1 (and where = ( z : ' z l < I 0 1 Let x ( z ) D e t n e c h a r a c t e r i s t i c function of A; then x i s a polynomial X(z) = l " = 0 a j Y j ( z ) -We know that the sequence of polynomials Xn> n >^ 0 given by t . , nL(9) v X n ( z ) = X(x 'z) obeys the conditions of Theorem U i n §11.1.3 with respect to E. We ft define the sequence X n' o n G by Y^* = x n(<()~ 1(z)) , and f i n d that Y , n > 0 f u l f i l l s the conditions of Theorem U with A n — respect to E(9,I) = <{>(E). (This i s seen by considering the dual mapping of <J> from k[x] to k[x] : we have for each integer j >. 0 an integer j * f o r which Yj*(z) = Yj (4>~1 ( z)) > z 6 G . We note that j * -»• °° i f and only i f j -> °° ; s e e also below, i n §2.4.) This completes the proof of Theorem 3. §2.4 A more general problem. We enunciate i n general terms the problem which i s to be our main preoccupation throughout the rest of this chapter. 69 Let ••• be a sequence i n k{x s a t i s f y i n g IE I > 1 for each j , and l e t 1^, 1^, ••• be a sequence of subsets of k[x] such that max I e ( E , • • -E 1 0 as i <=° E £ I . 1 i (the round brackets here do not denote f r a c t i o n a l p arts). Define the compact subset E of k{x by (59) E = { Z e. ( ? . . . . C J " 1 : £, £ i j . i = i i l l i i We s h a l l only be concerned with cases where E <= G. The question i s : when i s E a set of uniqueness for G? (In fact, can a simple c r i t e r i o n b given for even deciding i f E has measure 0?) Of course, we s h a l l not b answering these questions completely here. We discuss two cases which ar i s e . Case I. The 'no overlap' case. (This terminology w i l l be j u s t i f i e d below, i n the discussion following the statement of Proposition 10.) We suppose f i r s t that each I i n (59) s a t i s f i e s x i c W L ( E . ) - i = { p £ k [ x ] : l p l < l ? i l } -Let N. = L ( E . ) for each i >_ 1, and define the sequence m ,- k >^  0 by m^  = 0 and m^  = N x + N 2 + ... + N k , k > 1. Every z £ G can be written uniquely i n the form z = Z p x i i=l i ' where p^ £ W f l . - 1 f o r e a c n 1 >. 1 • The mapping cp given by (60) cp : Z p x m i - Z p. ( E , • • -E •) i = l i i = i l l l 70 is a homeomorphism and a group isomorphism of G onto G. cj> gives r i s e to various isomorphisms, and we can normalize our problem here as follows: PROPOSITION 10. In the non-overlap case j u s t described, the set E i n (59) i s a U-set for G i f and only i f E* i s a U-set, where (61) E* = {1° e . x " m i : e. £ I.}. i=l i I I This result p r a c t i c a l l y 'includes' Theorem 3 on page 64; i t certainly helps make i t more transparent. The set E(9,I) i n Theorem 3 i s the set (59) with a l l £ = 9 , and each 1^ some fixed proper subset I of WT,Q. .. The set E i n (61) becomes E* = {Z e.x K ': e, £ 1} L(.y;-i i=l 1 x i n this case, evidently a set of the type with respect to the seq-L(9) 2L(9) uence 1, x , x The elements e^, G^, ••• can be thought of as representing independent and non-overlapping blocks of powers i n * -1 the development of a z £ E in powers of x , and with I not being a l l of W , Q. 1 , there i s at least one sequence of length L(9) which i s L(.H; - i 'forbidden' i n any of these blocks. We leave i t for the reader to prove Proposition 10; we do not have room here for the d e t a i l s . We note that the dual mapping of cj> i n (60) leaves invariant the blocks of integers of the form { j : p mk <^  j < p mk+l} = B ^ , k ^ 0, and merely permutes the integers of any given block. One sees that PF(E) = {0} for E i n (59) i f and only i f PF(E*) = {0}. In §11.4 we w i l l see examples of perfect M^-sets of measure 0 defined by the non-overlap case. In fact, (59) always yields an M^-set i f l l ^ l / IC^I ->- 1 as i ->- °°, where 11 I denotes the c a r d i n a l i t y of I N • ( \E,A = p x i s of course the c a r d i n a l i t y of ^ j j - 1' w n ^ c n contains I_^  71 by assumption since we are i n the non-overlap case; thus the f r a c t i o n cannot exceed 1). In terms of (60), E i n (61) i s always an M^-set —N • ft whenever 11^ I p 1 1. It i s easy to construct such sets E so that they have measure 0; this w i l l be done i n §11.4. This i s one extreme. ft At the other extreme, such as i n Theorem 3, we have that E i s always a U-set i f l l . l i s bounded, and I. i s a proper subset of W„ . i n f -l l N^-l i n i t e l y often. PROBLEM. Find the dividing l i n e between U-sets and M-sets for the sets E* i n (61) i n the non-overlap case. We expect, for example, that some simple condition on the growth of '1-;' with respect to the sequence nu w i l l y i e l d E* as an 'u l t r a - t h i n ' set which i s always a U-set for G. This problem must be eminently tractable, with independent random variables at play (cf. chap. VIII i n [28]). 72 Case I I . 'Overlapping' allowed. We consider sets E as i n (59), but we no longer assume, as we did in Case I, that I. c W„ . for every i . Here the problem i s more subtle, i N-J-1 EXAMPLE. Let I = { 0, l,x} and consider 0 0 - i E = { I e.x : e. G I}. 1=2 1 i The subscript 2 on the sum ensures that E c G. If p = 2, then E i s a l l of the dyadic group G , so assume that our prime here i s p > 3. It i s easy to see that no member z G E can have two consecutive 2's as c o e f f i c i e n t s i n i t s standard development -1 -2 z = a.x + a„x + ... , where 0 < a. < p for every i . Put another 1 2 — I n n 1 2 way, the f r a c t i o n a l parts of x z s a t i s f y (x z) £ 2x + 2x + for a l l z G E and n = 1, 2, ... ( G^ i s the subgroup {z: L(z) < -2}, as usual). Thus E i s a simple set of the type H ^ , hence i t i s a U-set, by Proposition 8 on page 62. What happens i f we consider, instead of the set E just given, the set E(6) defined by (62) E(8) = { I ~ = 2 e i0" i: e ± G 1} , where 161 = p ( i . e . , L(6) = 1) and I = {0, l,x} as before? In the non-overlap case, this substitution would not affect the U-set status of E , as we have seen. But i n §11.6 we w i l l see that E(6) i s a U-set i f and only i f 8 i s a Pisot or Salem element (at least i n a l l cases where p > 3 ; we are not sure about the case p = 3). 73 Notes. The sets E in (61) which arise i n the non-overlap case are e a s i l y 'visualized' in terms of dissections of the unit i n t e r v a l (carried out i n the f a m i l i a r Cantor manner). Starting with E^= [0,1], we break E Q up into the p N* i n t e r v a l s of the form [jp N l , ( j + l ) p N l ] , where 0 <_ j < p ^ , and choose a union of l l ^ i of them to form our set E^. With E^ chosen, and consisting of 11^I...11^1 i n t e r v a l s of length p~ mk each, we form E ^ + ^ as follows. Each i n t e r v a l in E^ l i t into p^k+1 i n t e r v a l s of length p mk+l , and 11^.-^' °^ t n e m i s sp are chosen to remain i n E, ,.. This i s done i n the same way for each k+1 i n t e r v a l i n E^, so that a l l of the i n t e r v a l s i n E^ 'look' exactly the same after dissection. The sets E(8) given by (55) in Corollary 5, namely E(9) = {X E.8" 1: e. = 0 or 1}, i=l i I correspond, i n the case of 8 = x , to the Cantor sets C(p ) which we discussed i n our Introduction (here, we assume the p o s i t i v e integer r i s 21 2 when our prime i s p = 2) . The mapping (3) on page 10 maps r r E(x ) onto C(p ). As far as the case p = 2 i s concerned, the sets C(2 ) were known by Sneider [35] to be U-sets for Walsh series on the unit i n t e r v a l . The set C(3) i s the f a m i l i a r ternary set, but const-ructed on the i n t e r v a l [0,%]. Now C(3) corresponds to a U-set i n G for the case of p = 3 , and i t would be very i n t e r e s t i n g to discover whether or not C(3) i s a U-set for Walsh se r i e s . This problem seems intractable at the moment. Corollary 5 was obtained by Yoneda [43] i n the Walsh case of p = 2. We obtained the result independently, having been a long time ba f f l e d by the fact that none of the natural 'Lebesgue singular' measures sup-ported on the sets E(8) i n (55) have t h e i r F o u r i e r - S t i e l t j e s trans-forms vanishing at i n f i n i t y . 74 §3. M-sets of measure 0 v i a Riesz products. So far we have given examples of perfect sets of uniqueness f o r G (G = {z: L(z) < 0}), but we have provided no evidence that M-sets of measure 0 act u a l l y e x i s t . In th i s short section we use a f a m i l i a r object, the Riesz product, to give examples. No o r i g i n a l i t y i s claimed, but we cannot leave out t h i s elegant method. Things are p a r t i c u l a r l y simple here; events, characters and so on assoc-iated with a general I.e.a. group which appear or behave as i f they are 'independent', very often a c t u a l l y are independent when the group happens to be our group G. For s i m p l i c i t y , we do not aim for any generality at a l l i n Proposition 11 below; rather the reverse. In addition to the martingale convergence theorem of §1.6, we s h a l l use the following well-known r e s u l t from the theory of p r o b a b i l i t y : Let X^, ... be a sequence of i . i . d . r e a l random variables with means 0 and variance 1. Let a^, a^, ... be a sequence of CO r e a l numbers. Then the serie s Z^ converges a.e. or diverges 00 2 0 0 2 a.e. according to whether or not I, a. < <». I f Z, a. = °° then 1 1 1 1 both lim i n f S = -°° and lim sup S = °° hold a.e., where n-*-00 n n-»-00 n S = Z n . a. X. . n i = l 1 1 Now for any r e a l sequence a^, a^, a^, ... , a character series S on G i s well-defined by s e t t i n g 00 (63) S = n ( 1 + a Y k + a n ) • k=0 p P 75 The p N - t h p a r t i a l sum of S i n (63) i s given by N-1 (64) S N ( z ) = n.-( 1 + 2 a R e ( v k ) ) , N > 1 . P k=0 k P PROPOSITION 11. Suppose \a^\ < h for each k > 0, l i n ^ = 0, o and Z^ a^ = °° . Then the series S defined by (63) converges to 0 almost everywhere on G : lim S (z) = 0 a.e. . J n n Proof of Proposition 11. The p a r t i a l sums S M of S i n (64) are c p nonnegative, and / S N ( z ) m(dz) = 1 for each N > 0. Thus the sequence G P 1 {Spjj} forms a martingale which i s bounded i n L , hence i t converges a.e. by the martingale convergence theorem. Let us i d e n t i f y i t s l i m i t . We have N-1 (65) 0 < S p N < exp( 2 Z a kRe(y pk) ) for a l l N >_ 1, since 1 + x < e for a l l r e a l x. Since {Re(y k) ^  i s a sequence of i . i . d . random variables of P - 2 N _ 1 mean 0, and since Z a, = 0 0, we know that lim i n f Z a Re(y fc) = k N.+ co k=0 k P - 0 0 almost everywhere on G . I t follows that S N 0 a.e. on G . Fix any Z Q £ G for which S pN(z^) ->- 0. We w i l l see that one must also have S (z_) •+ 0 as n -> 0 0 for the f u l l sequence of p a r t i a l n 0 sums at z . By considering the p a r t i a l sums S (z n) within blocks of U n u N N the form rp < n < (r+l)p , where N ^  0 and 1 _< r < p, i t i s quite easy to show N N+1 (66) S n ( z Q ) = S p N ( z Q ) + 0(3^) max N I S . ( z 0 ) | , p < n <p 0< j<p 76 (In Che case p = 2, one has S 2 N +_.(z Q) = S 2 N ( Z q ) + a ^ Y ^ ^ g ) S j ( z ( p for N > 0 and 0 < j < 2 N .) Since S ^ N ^ Q ) -*• 0 and •+ 0 , one sees by induction that the sequence S ^ z ^ ) , n :> 0 , i s f i r s t of a l l bounded, and once this i s known, (66) shows also that i n fact S (z.) ->• 0 . n U This completes the proof of Proposition 11. Now l e t E be the set of points i n G where the series S f a i l s to converge to 0. E i s an M-set of measure 0. What i s the nature of E ? PROPOSITION 12. The M-set E of measure 0 associated with the series S i n Proposition 11 i s residual i n G; that i s , i t s complement i s a set of the f i r s t Baire category. Proof of Proposition 12. The set of points z € G for which the sequence ( S pN(z)} i s unbounded i s a G^-set i n G (a countable i n t e r -section of open s e t s ) . It i s easy to see from (64) that the set of such z i s dense i n G , thus E contains a dense G r-set and hence E i s o re s i d u a l . Remark (in connection with the proof just given). I t i s i n fact generally true that i f S i s some series whose p N - t h p a r t i a l sums S P N are everywhere pointwise bounded on some basic open set I , and i f 0 a.e. on I , then S P N eventually vanishes on I (indeed, under these conditions one would have S w = 0 on I for a l l m > 0, mp^ o — i f I i s a coset of G say, where NQ :> 0). This follows, for example, 77 from the main r e s u l t i n [41 ], which we mention i n §11.5. Evidently, none of the p a r t i a l sums i n (64) vanishes anywhere on G when Ia^I < h for a l l k . Notes. In the case of p = 2 , the M-sets considered i n t h i s section have been studied by Sneider [35J and Coury [11]. An argument s i m i l a r to that used i n the l a t t e r part of the proof of Proposition 11 can be found i n the proof of Th.(7.7) on p.209 of [46, vol.1 ]. We are indebted to Professor Coury for showing us t h i s argument, and f o r pointing out Propo-s i t i o n 12 to us. It i s shown i n [33] that the serie s S considered i n Proposition 11 for the case p = 2 have a l l of t h e i r p a r t i a l sums nonnegative, but we do not know i f the p a r t i c u l a r s e r i e s we have considered here have t h i s property when p > 2. The f i r s t example of an M-set of measure 0 for trigonometric ser i e s on the c i r c l e was given by Men'shov i n 1916. ooOoo §4. A class of perfect M^-sets of measure 0. The M-sets constructed i n the previous section are a l l 'large' i n the sense of Baire category. We return to the discussion which we began i n §11.2.4 on c e r t a i n perfect subsets of G = {z: L(z) < 0} , and show 78 that not a l l of the sets of measure 0 which arise i n the 'non-overlap' case are sets of uniqueness for G. THEOREM 4. Let N^, ^ , ... be a sequence of p o s i t i v e integers. Let nig = 0, and for k = 1, 2,3, ... l e t m^  = + . . . + N^. Suppose that for each i > 1, I. i s a subset of W N . _ 1 = { p £ k [ x ] : L(p) < N.^ }, and l e t E be the compact subset of G defined by (67) E = { E°° e:. x " m i : e. £ I.}. i=l i i i N • If lim I I I / p 1 = 1 , then E i s an M-set. i co Here, 11.I denotes the c a r d i n a l i t y of I. , which i s , of course, I I N • — ' W N - - l ' = P 1 • The normalization given by Proposition 10 in §11.2.4 (page 70) should be r e c a l l e d at this point: each term x m i in the sum (67) can be replaced by an arb i t r a r y element £ k{x ^}, provided In I = p m i , without aff e c t i n g the r e s u l t . Now Theorem 4 i s i n t e r e s t i n g only in the case where the set E i n (67) CO _ x j has Haar measure 0, and this i s true i f and only i f II 11 |p i = 0. i=l 1 79 It i s easy to f i n d examples where this holds, along with the condition -N • i n the theorem that II. I p 1 1. Notice that this l a s t condition nec-l essitates then that N. -> 0 0 as i 0 0. x Proof of Theorem 4. Regard z^, ... as a sequence of independ-ent random variables, with each z^ uniformly d i s t r i b u t e d on i t s range 1^ , and l e t y be the pro b a b i l i t y measure on G which represents the d i s t r i b u t i o n of the random variable CO _ - m,-Z = I Z . X i . 1 1 Then y i s supported i n E. We w i l l show that u(n) -»• 0 as n -*- 0 0 ; then the result w i l l follow from Proposition 6 i n §11.1.1. Now i f f(z) i s a fi n i t e - v a l u e d function on G which, when r e s t r i c t -ed to E , depends only on f ° r some fixed k > 1, then one has / f(z)u(dz) = (1/11,1)1 f ( e , x " m k ) . G K e k t lk * m,. k - l V Suppose k > 1, and assume p < n < p K m, Then we can write n = up ^ 1 + h, where 1 <^  u < p ^ and m k - l 0 _< h < p , and we have Y n = Y m-k_l = Y m k_i Y n up + h up K \ Now y m._, depends only on e. , while y depends at most on up x k h e l ' - ' * ' e k - l * These two functions are therefore ' y-independent random variables' on E , and hence we can write 80 y(n) = / Y dy = J Y m k_idy / Y dy _ n r, U p r 11 I t follows that I y(n) I < I / Y mk-l dV Q UP I I Y m k_ 1<e k*" m k> I e, £ I, U P II. I " k " k k 1 I I Y ( e v x Nk) I. E, £ I. u II, I "k" ^k Since 1 < u < p N k , we have I Y ( N k ) = 0 , and so y(n)| < J_ |WNk_1 v i k | P % / H k l " 1 By hypothesis, t h i s l a s t term tends to 0 as k 0 0 , and hence y(n) -*• 0 as n -»• 0 0 . This completes the proof of Theorem 4. Remarks. Afte r discovering the M-sets i n Theorem 4, we learned that Skvorcov L34] had previously discovered a class of perfect M-sets of measure 0 on the unit i n t e r v a l for Walsh se r i e s . We are not sure what relationship our sets have with his i n the case p = 2 . *** §5. N u l l series with slow growth - and a theorem of Wade and Yoneda. N The 'growth' referred to here i s that of the p -th p a r t i a l sums of a series. Wade has shown i n [37] that i f S i s some series on G having the two properties: (i) S M (z) ->- f(z) outside a countable set, P -N ( i i ) p S M ( z ) ->• 0 everywhere on G , where f i s an integrable function, then S must be the Fourier series of f . Consider the case where S i s the D i r i c h l e t kernel D on G : D = I y • 0 n N -1 Now D w = p 1 G , where G. = { z£ k{x }: L(z) < -N } ; thus p N IN 82 -N D M(Z ) -»- 0 everywhere save at z = 0, and p D M f a i l s only at p N P z = 0 to converge to zero (at zero i t i s i d e n t i c a l l y 1 ) . Hence the condition ( i i ) above cannot be relaxed by dropping even a single point. How far can condition (i) be relaxed? It cannot be relaxed to 'con-vergence a.e. ' ; we have already seen examples of n u l l series S which converge a.e. to 0 without being i d e n t i c a l l y 0. The c o e f f i c i e n t s of such S have to tend to 0, so these S cer t a i n l y f u l f i l l condition ( i i ) . Let us turn things around a l i t t l e , and assume S i s some series with the property that S M 0 a.e.. What growth condition stronger P than ( i i ) for z i n the exceptional set w i l l guarantee that S must be the zero series? Wade and Yoneda [41] proved the following r e s u l t : assume there i s a sequence N < N„ < such that (a) S ->• 0 a.e. on G , (b) lim sup |S w.l < P J CO everywhere on G ; then S must be the zero se r i e s . Note that Chow's martingale convergence theorem (see §1.6) implies that the sequence {S M.} must i n fact converge a.e. for any series P 3 sa t i s f y i n g (b). The purpose of this section i s to show that the res u l t of Wade and Yoneda i s sharp. If S i s some nonzero series s a t i s f y i n g (a), then 83 c e r t a i n l y sup I S ^ I -* 0 0 as j - * - 0 0 (as Lebesgue's dominated conver-G P J gence theorem would show), but these suprema may become unbounded a r b i -t r a r i l y s low ly . THEOREM 5. Let 1 = < < . . . be an a r b i t r a r y inc reas ing sequence of r e a l numbers tending to 0 0 . Then there e x i s t s a nonzero s e r i e s S on G s a t i s f y i n g the two condi t ions (a 1 ) l im S (z) = 0 everywhere of f a per fect set of measure 0, (b 1) sup IS „ I < cx , N = 1 , 2 , 3 , G P N Proof . There i s no loss of genera l i t y i n assuming that a,_^ / a , 1 as k °° . Choose a sequence of p o s i t i v e in tegers < N < . . . fo r Theorem 4 i n the previous s e c t i o n (page 78 ) , and for d e f i n i t e n e s s , l e t us assume that the sets I, cz VJ , i n that theorem have been chosen k Nfc-1 so that t h e i r c a r d i n a l i t i e s obey (68) 11,1 = [ < a k _ l / a k ) p N k ] + l * k - 1 -Let E be the per fect set given in (67), and l e t y be the n a t u r a l p r o b a b i l i t y measure c a r r i e d by E (as per the proof of Theorem 4) . By Theorem 4 E i s an M - s e t , because 11, I p N ^ -> 1 as k ->- 0 0 . 84 The transform y(n) of u vanishes at °° , and the Fourier se r i e s 00 ^ of y , namely S = S[y] = I Q y ( n ) Y n , converges to 0 everywhere outside E . The Haar measure of E i s 00 _ N m(E) = n I I. I P k = 0 k = l K as can be seen from the estimate k (69) 1 K. M 1 — N i < n I I . I p ~ N i < l e x p ( I (a. / a. .)p N i ) , k > l , i -1 = 1 . 1 1— 1 a, i = l a, -1- 1 Fix k > 1 , and assume A i s some coset i n G of G m k I f y(A) i s not 0 , then y(A) = n I I . I ' 1 = p" mk n p N i I I . I " 1 i = l 1 i = l 1 < a, p m k a. m(A) From t h i s i t follows that y(A) < a^m(A) for any A i n the f i e l d of sets generated by cosets of G m . k Suppose N _< m^  , l e t z Q € G , and l e t A = ^ ( z ^ ) denote that t of G.T which contains z„ . N 0 With S = S[y] we deduce, with the help of (49) i n §11.1.2 (page 54) that 0 < S n N ( z n ) = p y(A) N < p a k m(A) = a k , and Theorem 5 i s proved Note that , for every z £ E , S pm k( z) i - s the r e c i p r o c a l of the 85 centre term i n (69) for each k :> 0 . We have not looked at the f u l l sequence ^8 } of the p a r t i a l sums of S here. C e r t a i n l y sup IS I -»• °° as n -* °° , f o r i f th i s were not G n the case we should have a uniformly bounded subsequence {S n } , which j would imply 1 = u(G) = constant term of S = / S n, dm + 0 as j + °° . G 3 ooOoo §6. Uniqueness and members of T - r e s u l t s f o r sets E(9,I). We have already seen evidence that whether or not a subset E of G = {z: L(z) < 0} i s a set of uniqueness f o r G depends on the a r i t h -metical structure of E . The theorem that follows deals with a class of sets constructed with 'overlap allowed' ( case II of §11.2.4; see page 72). 86 THEOREM 6. Let 9 £ k{x~ } s a t i s f y 161 > 1 , but suppose L(8) > 1 i f p = 3, and L(6) > 2 i f p = 2. Let I = 1(6) = { 0, l,.x, . .. , X 2 L ( 9 ) - 1 } , and l e t E = E(6,I) be the perfect subset of G given by ( 7 0 ) E = { I e.8 - 1 : e. G I }. i=2 1 1 Then E has Haar measure 0 , and E i s a set of uniqueness precisely when 6 belongs to T. The subscript 2 on the sum in ( 7 0 ) ensures that E c G . DISCUSSION. The proof of Theorem 6 w i l l be given i n two parts, Theorems 7 and 8 below. Theorem 7 w i l l deal with the case ' 6 £ T ' , and Theorem 8 with the case when 9 £ T . As w i l l be seen, Theorem 6 does not t e l l the whole story ( but nor do Theorems 7 and 8 either, for that matter!). The requirement that 6 be a ' l i t t l e b i t larger' when our prime p has the value p = 2 or 3 i n Theorem 6 i s there to ensure that the c a r d i n a l i t y of I s a t i s f i e s I I I < |9|. We w i l l see i n a moment that this condition guarantees m(E) = 0 i n ( 7 0 ) . Before that, however, l e t us present the 'two halves of the p r o o f of Theorem 6 i n a condensed form, and also discuss some examples. PROOF OF THEOREM 6 (CONDENSED). 9 H. T. E i n ( 7 0 ) carries a natural probability measure y. Look at i t s Fourier transform; i t vanishes at °° , so that Propos-i t i o n 6 i n §11.1.1 applies. Key to the observation that y(n) -»- 0 i s Grandet-Hugot' s theorem 87 in §1.3 (page 22), which characterizes the members of T; there i s no nonzero A. £ k{x }^ for which the f r a c t i o n a l parts of A8 s a t i s f y | ( A 8 n ) | < 181 for a l l s u f f i c i e n t l y large n. 8 £ T. Let s be the algebraic degree of 8 . Then E i n (70) (s) i s a set of the type H , so that Proposition 8 in §11.2.2 applies. (s) Key to our observation here, that E has type H , i s Proposition 4 in §1.5.2 (page 31). There exists a nonzero A , and one not too large, such that a l l of the f r a c t i o n a l parts ( A 8 n ) are small (as small as we l i k e ) , n = 0,1, 2, .... The proof uses also the c a r d i n a l i t y condition III < 181 , which implies by i t s e l f that E has measure 0 as already mentioned. EXAMPLES. 8 = x i s a Pisot element (the simplest Pisot element of a l l ) , and the set °° - i E = { Z e.x : e. = 0, 1, or x } 2 i i i s a U-set for G whenever the prime i s p > 2. If p = 2 i n this example, then E i s a l l of G . The case p = 3 doesn't follow from Theorem 6, however ; moreover, the c a r d i n a l i t y of I = {0, 1, x} i s III = 3 = 181 when p = 3. We dealt with this case though i n §11.2.4; see the example on page 72. 8 = x + x i s not a member of T , since i t belongs to k(x) and i s not i n t e g r a l . Accordingly, the set E = { I e.(x + x ) 1 : £. = 0, 1, or x } 2 1 1 i s not a U-set for G (whatever the prime p; see Theorem 7). 88 QUESTION. Let 6 £ k{x } be any element s a t i s f y i n g L (9 ) = 1 ; thus 9 = a ^ + a Q + a ^ x - 1 + Let I = {0, 1, x} , as i n Theorem 6 ( 2L(0) - 1 = 1 here ) , and l e t E be the set given i n (70), namely E (9,I) = { I e . 9 " 1 : e. £ I>. i=2 1 1 The question i s : what happens when p = 3 ? When the prime i s p > 3 , Theorem 6 gives the answer; E (9,I) i s a U-set here i f and only i f 9 belongs to T. When the prime i s p = 2, E(6,I) i s a l l of the group G (certainly not a U-set!). This leaves p = 3 . Suppose p = 3 then. Now Theorem 7 w i l l show that E (9,I) i s an M^-set i f 0 does not belong to T, but we do not know i f this set has measure 0 any more (we have I I I = 191 = 3 again!). So what i s the measure of this set? Leave p = 3 . In the example on page 72 we chose 9 = x and showed that the set was of type H ^ , hence a U-set. Other 9 ? ANOTHER QUESTION. As we have seen ( §11.2 Notes; page 73), non-overlap type sets may be v i s u a l i s e d on the unit i n t e r v a l . Is there a way of picturing the sets E given i n (70) too? These sets have turned out to be interesting a l g e b r a i c a l l y , so i t would be good to have a geo-metric picture of them as well, i f i t ' s possible. 89 PROOF THAT III < 181 IMPLIES E(8,I) HAS MEASURE 0. (Thus a l l of the sets E i n (70) of Theorem 6 have m(E) = 0. The condition 1 III < 161 1 i s not a necessary condition for measure 0, however, as the example on page 72 shows.) Consider any 8 € k{x l} for which 181 > 1 , and suppose I i s a f i n i t e subset of k[x]. Let r = max L(e) , and consider eGI OO F = { I e . 9 1 : e . £ I} . i - 1 x x We can write, for N > 0, N °° F = { I £ . 8 1: £. e 1} + { I £ . 8 x : £ . £ 1} 1=1 i=N+l = Q N + H N , say. If z £ H N , then L(z) < r - (N+1)L(8) ; hence H^ c G^, , where N 1 = (N+1)L(8) - r - 1 , and G , = {z: L(z) < -N1} (as usual). Thus m(H ) < p r + l-(N+l)L(9) , , N Now has not more than HI points, so we obtain m(F) < I I| Nm(H N) < p r + 1 " L ( 0 ) ( 111 / | 0 | ) N . P l a i n l y , i f III < 191 , then we must have m(F) = 0. Q.E.D. 90 We r e c a l l the meaning of W for r > 0: W = {to £ k[x ] : L(to) < r}. r — r — THEOREM 7. Let 9 € k{x suppose 101 > 1 , and assume that 9 does not belong to T. Let I be a f i n i t e subset of k[x], and l e t E be the compact subset of k{x given by C O (71) E = { I e.9" 1: e, E l } . i = l 1 1 If I includes 0 together with a basis B for the ( f i n i t e ) vector 2 -s space (over k) W , Q. = {ai £ k[x]: Itol < 191 } , or i f x I con-£L(v)— 1 tains 0 and such a B for some integer s ^  0 , then E i s an M^-set; that i s , E carries a measure y * 0 whose transform vanishes at 0 0 . This r e s u l t could be refined somewhat, but i t i s s u f f i c i e n t for our purposes here. Let us see how Theorem 7 implies that the set (70) i n Theorem 6 i s an Mg-set for G i f 9 ? T . F i r s t note that the set I = { 0, 1, x,"..., x 2 ^ 0 ) - 1 } appearing i n Theorem 6 contains 0 and a simple basis for W , Q. there. ZL(.fc); —1 Suppose we have a measure y * 0 carried by the set E i n (71) with * -1 the property that y(u) •+ 0 as lu| -»• °° i n k{x }. Choose any k > 0 -k such that the set E' = 9 E i s contained i n G. I f we define the measure y' on G by setting y*(A) = y(0 kA) for Borel sets A in G , we find that y' i s carried by E' , and y'(u) -> 0 as |u| -v 0 ° as well. 91 Proof of Theorem 7. For s i m p l i c i t y i n the proof, l e t us assume that I contains {0} U B , where B i s some basis f or W . . (the other ZL(,o; —1 p o s s i b i l i t i e s f o r I are dealt with s i m i l a r l y ) . Let I Q = {0} U B , and l e t be the compact subset of E : CO (72) E q - { i e . e " 1 : a y . 1 = 1 Regard the , i _> 1, i n (72) as a sequence of i . i . d . random va r i a b l e s , each one uniformly d i s t r i b u t e d on i t s range 1^ , and l e t y be the p r o b a b i l i t y measure on k{x l ] representing the d i s t r i b u t i o n of the random v a r i a b l e z = I e.9 x . i=l i Then y i s ca r r i e d by E^ , and we have c E. ( More formally, i f you l i k e , give 1^ the d i s c r e t e topology, and assign mass ( 2L(9) +1) 1 to each of i t s points. Then the obvious map oo from the compact measure space II In onto E„ i s continuous, and i t i=l 0 0 ' gives r i s e to the d i s t r i b u t i o n y on E Q - ) Define cp(z) = ( 2 L ( 9 ) + 1 ) _ 1 ( 1 + I Y ( z ) ) , z e k{x - 1}. e€B e There i s a constant T < 1 such that ( c f . §1.5.1) (73) cp(z) = 1 i f |(z ) | < |9|" 2 , | cp(z)| < x otherwise. Denoting the transform y(u) by a(u) a(u) = y(u) = / Y (z)y(dz) , u £ k{x 1}, -1 u k{x l ] 92 we compute a(u) = n ((2L(6) + 1)" I Y (e8 n ) = n <|)(u8 n ) . n = l V e G I 0 u 1 n-1 From t h i s we obtain, f o r any A £ k{x 1}, m-1 (74) a ( A 9 m ) = a ( A ) II 4 > U 9 J ) , m > l . 3=0 Since 8 € T , Grandet-Hugot's theorem ( §1.3, page 22 ) and (73) show that, f o r every A * 0 , I <j>( XQ2 ) | < T < 1 for i n f i n i t e l y many j >_ 0 . It follows from (74) then, that |a( A 9 m ) I decreases to 0 as m -*- » for a l l A * 0 , and since a i s continuous, t h i s convergence i s uniform on the compact set {A: 1<^|AI < 191}. Since an a r b i t r a r y u * 0 can be written i n the form u = A 6 m , where 1 _< IAI < 191 , we conclude that a(u) -»• 0 as |u| 0 0 . This completes the proof of Theorem 7. And now for.the other h a l f of Theorem 6. The sets i n (70) a l l carry natural p r o b a b i l i t y measures y , but i f 9 belongs to T then y cannot vanish at °° (the way i t j u s t did for 9 t T) , as we now show. THEOREM 8. Let 9 belong to T , and l e t I be any f i n i t e subset of k[x] with c a r d i n a l i t y III < 161 . Then the compact subset of k{x }^ given by CO (75) E = { I e.9" 1 : e. £ I } i=m 0 93 i s a set of uniqueness for G , where i s any integer large enough to ensure that E c G . Proof of Theorem 8. Let s >^  1 be the degree of 0 over k(x). (s) We w i l l show that E i n (75) i s a set of the type H , so that Prop-o s i t i o n 8 i n §11.2.2 w i l l y i e l d the r e s u l t . Now i t i s readily seen that the vector sequence given by (76) V ( m ) = ( [A0 m], [A 0 m + s~ 1] ) , m > 0 i s normal for any X * 0 i n k{x (see §11.2.2; page 62). (s) We w i l l f i n d a X * 0 so that E becomes an H - s e t with respect to V ( m ) . For a positive integer N (to be s p e c i f i e d l a t e r ) , l e t us use Propo-s i t i o n 4 i n §1.5.2(page 31) to choose X * 0 s a t i s f y i n g (77) |AI < p ( N + " o U 9 ) ) ( s - l ) ( and (78) l ( A 0 j ) | < p - ( N + m o L ( 9 » f o r A L L ^ Q , (On the l e f t i n (78), we are dealing i n f r a c t i o n a l parts) oo For z = I e. Q~3 E E , we have for m > mn , J — 0 (79) A0 m z = I e .A0J + I e_^.XQ~3 . j=0 m - J j = l Note that the assumption that E l i e s i n G implies L(e) < m QL(0) for each e € I. 94 This and (78) show that the f r a c t i o n a l part of the f i r s t sum i n (79) belongs always to G M ( i . e . , to {u: L(u) < -N}). Now note that f o r e € I we have (using (77)) L ( e A e ~ J ) < m L(9) + (N+m 0L(8))( s-1 ) - j L(6) , and hence (80) L( eAe~ J) < -N i f j > J = [ N s / L ( 8 ) ] + B , where the i n t e g e r B i s independent of N . I t f o l l o w s from (78), (79) and (80) that we can w r i t e d = ([A8 m]z) m as d = ( I e . \ Q~2 ) + u (z) , where u (z) £ G., . m N g Now the number of open rectangles A c G which are products of Ns cosets i n G of G„ i s p , and to i u s t which one the v e c t o r N ' J V U ) ( z ) = ( d d ) m m+s-1 belongs i s completely determined by the sequence e e m+l m+J+s Hence, as z ranges over E and m over the p o s i t i v e i n t e g e r s , the v e c t o r V ^ m \ z ) cannot p o s s i b l y meet more than | I | ^ + S of these re c t a n g l e s A . Since III < |0| by assumption, we know that III = c 181 f o r some constant c < 1. Thus we have H | J + S = c J + s p L ( 8 ) ( J + s ) 95 and a glance at the expression f o r J i n (80) shows that J+s L(9)(J+s) . Ns/L(9) Ns c p < A c p for a constant A > 0 independent of N . Hence a large enough N w i l l ensure that . .J+s Ns H I < P providing us with a A free of any vector V ^ m ^ ( z ) . This finishes the proof of Theorem 8. REMARK. Returning to Theorem 6 for a moment, l e t E(9) denote the set E in (70). If 9 i s not i n T , then E(9) i s an M-set, but i t does not remain so for small variations i n 9 since the Pisot elements are everywhere dense i n {z: |z| > l } . Contrast this with what happens on the c i r c l e for the sets C(9) discussed i n our Introduction. There, an M-set C(9) must remain an M-set for small changes i n 9 , since the Pisot numbers on R are closed. ooOoo §7. Some comments, with a question or two. Let us map everything over to the r e a l l i n e , so that the compact group G = {z e k{x 1}: L(z) < 0} maps to [0,1] . A l l of the perfect subsets of G map onto perfect subsets of the unit i n t e r v a l , and 96 any measures that they carry become measures supported by their images. If a given subset E of [0,1] i s known to be a trigonometric U-set or M-set, or a p-series U-set or M-set for some p a r t i c u l a r prime p, what can be said about i t s nature with respect to either the trigonometric system or di f f e r e n t primes p , as the case may be? This general problem must be d i f f i c u l t . What relationships exist between the r e a l f i e l d operations on [0,°°) and the p-series ones? There i s of course the i n t r i g u i n g question of whether or not the ternary set i s a U-set for Walsh series on the i n t e r v a l . Its resolution s t i l l seems a long way o f f . The sets considered i n Theorem 6 of the previous section are not ea s i l y v i s u a l i s e d when mapped onto the i n t e r v a l , but any of the perfect subsets of G constructed with 'no overlap' (in the sense of §11.2.4) can be easily pictured on the i n t e r v a l (cf. §11.2 Notes, page 73). Let E be one of the perfect M Q-sets f or G considered i n Theorem 4 of §11.4. Let E* be i t s image i n the i n t e r v a l , and l e t y* be the measure on E* corresponding to the natural measure y which i s carried by E. Is E* an M-set for trigonometric series? We can show that rl 2ninx . J e y (dx) -*• 0 0 for In I -»• oo along certain subsequences of the integers, but this i s not enough. If E i s some set of the type on G with respect to a seq-97 uence { x 1 1^ }^  k[x] , then i t s image i s an H -set on [0,1] with respect to the sequence {p™ 1} ; hence i t i s a trigonometric U-set as well. How far can observations l i k e this be extended, and w i l l a l l ' s u f f i c i e n t l y thin' subsets of G correspond to trigonometric U-sets on [0,1] ? How can we t e l l when a measure u has i t s F o u r i e r - S t i e l t j e s transform vanishing at 0 0 ? Progress has been made on this problem recently (see [26]). If y i s a p r o b a b i l i t y measure on the c i r c l e [0,1), then i t i s well known (see Th.(10.5) on p. 144 of [46, vol.II] ) that y(n) -»- 0 as |n| -»• 0 0 i f and only i f , for a l l subintervals I of (0,1), y({ x € ( 0 , l ) : (nx) £ l}) + m(I) as n ->- 0 0 , where (nx) i s the f r a c t i o n a l part of nx , and m i s Leb-esgue measure. Back on our group G , i t i s very easy to prove: PROPOSITION 13. Let y be a pseudomeasure on G , and suppose y(G) = 1. Then a necessary and s u f f i c i e n t condition i n order that y(n) ->• 0 as n ->- 0 0 i s that y({ z£ G: (u) z) £ 1}) -> m ( I ) n as n -»• 0 0 for a l l basic open sets I <=• G. (Here m i s Haar measure, and { con} i s our standard l i s t i n g of k[x] (as at (17) i n §1.2.2) .) This result can be written on the unit i n t e r v a l i n terms of the f r a c t -i o n a l parts (n-x), where n«x denotes the p-series m u l t i p l i c a t i o n of the natural number n with x £ [0,1). Let C(0) be one of the Cantor subsets of the c i r c l e alluded to 98 e a r l i e r : C(9) - { X e . 9 : e . = 0 or 1). i=l 1 1 Assume 6 i s not a P i s o t number. Then C(9) c a r r i e s a p r o b a b i l i t y measure y f o r which / e ^ ™ i j ( d x ) •> 0 as n °° . From t h i s , and 0 the comment made on the previous page concerning such a measure y (top d i s p l a y ) , we can i n f e r that / Y , ( n x ) P(dx) ->• 0 as n ->• <=° , 0 k where y i s any of our group characters considered on [0,1). N Since y (p x) = y M ( x ) i f we are cons ider ing the p - s e r i e s case , k k p w we have ~ N y ( k p ) -*• 0 as N + » fo r each f i x e d k , where the transform i s with respect to our group G . I t i s n a t u r a l to wonder here i f i n f a c t y(n) -*• 0 fo r n -»• °° a rb -i t r a r i l y , so that C(9) would become an M - s e t fo r G . 99 CHAPTER III SPECTRAL SYNTHESIS Among compact subsets E of k{x there are some that keep a very tight rein on the pseudomeasures that they support, obeying 'spectral synthesis' i n i t s strongest form. Certain of our compact sets E(9,I) do t h i s ; just which ones i s s t i l l a mystery, but the arithmetical structure of some gives the show away for them. The d e t a i l s are given i n §2 . Note. i n this chapter we are less expansive. The reader having found us perhaps too effusive i n the e a r l i e r chapters w i l l surely be compensated here, as we race along paths already mapped out by Herz and Meyer to see what happens with our sets E(8,I). §1. The problem of synthesis. Books have been written on this subject (e.g., [A]), and almost a l l modern accounts of harmonic analysis contain sections devoted to spectral synthesis (e.g., [22], [31]). The reader may wish to compare what we do i n this chapter with chapter VII in [281. For the moment, l e t G and V both denote our p-series f i e l d k{x and l e t A = A(T) denote the usual embedding of the convolution algebra L ^ G ) i n C Q(T) given by the Fourier transform. ( C ^ i s the space of continuous functions vanishing at i n f i n i t y . ) A i s a Banach algebra under the norm II f I I A = II f ll L l ( G ). 1 0 0 Let E be a closed subset of T , and l e t 1(E) denote the set of a l l g £ A which vanish on E . 1(E) i s a closed i d e a l i n A , and the zero set (spectrum) of 1(E) , namely, the set of points u . E T at which each g £ 1(E) vanishes, i s precisely E . Can there be some other closed i d e a l I i n A whose zero set i s exactly E ? Such an I must s a t i s f y J(E) c I c 1(E), where J(E) i s the closure i n A of the set of a l l g which vanish on some neighbourhood of E. 1(E) and J(E) are respectively the largest and the smallest of the closed ideals i n A having zero set equal to E . Whenever they coincide, we say that E obeys spectral synthesis (or that E i s a set of synthesis). When E i s a set of synthesis, each function f i n L^(G) whose transform vanishes on E can be approximated i n norm by functions f, IS. with transforms vanishing on neighbourhoods ft, °f E. (Let us mention here that whereas i t i s clear that any subset of a set of uniqueness i s again a set of uniqueness, the property of being a set of synthesis i s not hereditary i n this way.) In every nondiscrete I.e.a. group, t.here arc closed sets E which do not obey synthesis (Malliavin's theorem). Postponing i !,»- moment discussion of the best behaved sets, l e t us come at the general prob-lem with the Hahn-Banach theorems. The Banach space dual of A(T) i s the space PM = PM(D 101 of pseudomeasures on T ; PM i s isometrically isomorphic (under the 00 Fourier transform) with L (G). With the usual d e f i n i t i o n of the support of a pseudomeasure u , we write y £ PM(E) i f the support of y i s contained i n E (which means y annihilates a l l g £ A which are zero on some neighbourhood of E ; thus PM(E) i s the annihilator i n PM of the closed i d e a l J(E) ) . Naturally, a l l y i n the space M(E) of measures supported i n E belong to PM(E). Let M^(E) denote those measures carried by f i n i t e subsets of E, and l e t A(E) be the usual algebra of r e s t r i c t i o n s to E of functions i n A . A(E) i s a Banach algebra (isomorphic to the quotient algebra A / 1(E) ), and the norm i n A(E) i s given by II g l l A ( £ ) = i n f { || f || A : f = g on E } . Here then are three equivalent ways of asserting that a closed set E c T obeys synthesis: (a) E i s a set of synthesis; (b) The Banach space dual of A(E) i s PM(E); (c) M f(E) i s weak* dense i n PM(E) . Going on i n (c) i s the synthesis of pseudomeasures i n the weak* top-ology of PM (as dual of A ) , or, taking Fourier transforms, the synthesis of bounded functions with f i n i t e trigonometric sums. 102 From now on w e assume that E c: k{x }^ i s compact. A l l kinds of ident-i f i c a t i o n s can then take place, and we are e s s e n t i a l l y back to the s i t u -ation of dealing with absolutely convergent Fourier series and pseudo-measures as they were discussed i n §11.1.2. The algebra A(E) now has a unit, and the transform u of any y £ PM(E) i s a bounded continuous 'step-function' on k{x L}. Assume now that each y £ PM(E) i n (c) above i s the weak* l i m i t of a sequence of measures y , , k ^ 1 , i n M^(E). Then with a l i t t l e help from the Banach-Steinhaus theorem and Baire's theorem (cf. [28, p. 219]), we see that there must be some constant C (which cannot be less than 1) such that the following i s true: for each y £ PM(E) there i s a sequence y , , k ^ 1 , of (81) measures i n M^(E) s a t i s f y i n g II \ I I p M < C II y | | p M a n d y f c - y w e a k * . This i s the kind of strong synthesis we are interested i n . Note that II V a n d II V lljj» are the same thing for pseudomeasures V . The condition of strong synthesis asserted i n (81) can also be written as follows: for each bounded continuous function cp on k{x l ] whose 'spectrum' l i e s i n E , there i s a sequence cp , k >^  1 , of IS. f i n i t e trigonometric sums, whose frequencies belong to E , such that sup I cp I < C sup I cp I , and cp -> cp - l - i k k{x l ] k{x l ] uniformly on compact subsets of k{x 103 §2. Results for sets E(8,I). Recall that for each integer r >_ 0 , denotes the f i n i t e vector space (over our p-element f i e l d k) consisting of a l l polynomials i n k[x] having degree _< r : Wr = { w€ k[x] : L(u>) < r}. Let 0 £ k{x 1}, | 0 | > 1 , l e t I be a f i n i t e subset of k[x], and l e t E ( 0 , I ) be the compact subset of k{x }^ given by (82) E ( 0 , I ) = { I e. 0 _ I : e. E I }. 1=1 1 THEOREM S. If 0 i s arbitrary and I <z. WT / Q . , , or i f I i s arb-L ( U ; - l i t r a r y and 0 belongs to T ( the set of Pisot and Salem elements), then the compact set E = E ( 0 , I ) i s a set of synthesis. In both cases there i s a constant C for which (81) i s obeyed by the set E. In the former case C = 1; we also have C = 1 i n the l a t t e r case i f we assume 6 i s a Pisot element, and I contains {0} U B where B i s some basis for WL(9)-1-(An example of such a {0} U B is {0, l , x , x L ( 9 ) 104 This result could be refined, but b a s i c a l l y we must ask: what happens with a l l of the remaining sets E(9,I) i n (82)? Synthesis for a set E i s a very s p e c i a l property to have, especially the kind con-sidered here. In view of Th. VII on p. 255 of L281 » i t seems l i k e l y that a l l of the sets i n (82) w i l l contain compact subsets which do not obey synthesis at a l l . Now l e t us break Theorem S up into more manageable pieces, the proofs of which w i l l be sketched i n the sections which follow. F i r s t we state condition (81) i n terms of a sequence F , k _> 1, of f i n i t e subsets of E : for each y E PM(E) there i s a sequence y, , k >_ 1 , of measures i n M^(E) such that y, £ M ( F , ) for each k, (83) 11 M k "PM - C " U "PM a n d U k U weak*. THEOREM 9. Let 8 £ k{x _ 1} s a t i s f y 161 > 1, but be otherwise arb-i t r a r y , and assume I c W L ( Q )_ 1 = {co £ k[x]: |u| < 181 }. Then E = E(8,I) i n (82) obeys (83) with C = 1 and k F = { I E 6" 1 : E £ I } + A , * i = l 1 i k where A, i s a u y p 0 i n t of the set Q~k E for each k (e.g. A, = 0 i f 0 £ I ) . THEOREM 10. Let 8 belong to T , and l e t I be an arbi t r a r y f i n -i t e subset of k[x]. Then there exists a constant C, and a f i n i t e subset F of E = E(6,I) in (82), such that (83) holds for E with the sets k F, = { £ E 9 1 : E. e I } + 8 _ K F . k i = l 1 105 THEOREM 11. Let 0 be a Pisot element of k{x },. and suppose that the f i n i t e subset I of k[x] contains {0} U B , where B i s some basis for WT ... = { co£ k[x] : l<ol < 191 }. Then E = E(0,I) in (82) obeys (83) with C = 1 and k F = { I e. 9 1 : e. 6 I }. k . , I I i = l Note that i n each case (in the three theorems just presented), the f i n i t e sets F^ are contained i n the set E i n question. COMMENTS ON THE THREE PRECEDING THEOREMS. As 9 passes from being near arbitrary i n Theorem 9, to being a Pisot element i n Theorem 11, with the strongest synthesis ( C = 1) obtained for the sets E(9,I) i n these two theorems (note the opposite nature of the r e s t r i c t i o n s on I ) , there i s a 'middle ground' i n which strong synthesis i s s t i l l obtained with an arbi t r a r y I , provided 9 i s a Pisot or Salem element. The synthesis i n Theorem 10 i s a l i t t l e weaker than i t i s i n the other two theorems, but the constant C there c o u l d be estimated i n terms of I and 6. The e x i s t e n c e of this constant C (and the f i n i t e s e t F ) i n Theorem 10 comes from a general theorem of Meyer on harmonious sets of f r e q u e n c i e s (cf.§3; to follow). Theorem 9 comes ea s i l y with Herz' technique of d i s c r e t i z i n g a pseudo-measure; this w i l l also be elaborated on s l i g h t l y i n §3. It shows CO _ A strong synthesis for any set (I e.9 : e. = 0 or 1} where 191 > 1. —*- i=l i i Th eorem 11 ? That i s what most of the rest of this chapter i s a l l about, and Theorem 10 i s c r u c i a l to i t s proof (along w i t h Theorem 12 i n §4; 'Bochner's property'). 106 FURTHER COMMENTS. We make a few remarks for Che reader who i s perhaps fa m i l i a r with what happens i n the r e a l or p-adic cases (the r e a l case i s also discussed b r i e f l y i n our Introduction). Theorem 11 may be considered as the net result of Theorems 10 and 12; strong synth-esis with C = 1 when 9 i s a Pisot element (just as i n the r e a l case). But things are a l i t t l e b i t the 'other way round' here when compared with say, the r e a l results i n [28]. Inasmuch as 9 may be a Pisot or Salem element i n Theorem 10, we have more synthesis than i s known for the sets C(9) on the l i n e (or c i r c l e , i t doesn't matter). As i f to com-pensate for this however, we get 'less Bochner' than i s obtained i n [28] for these sets, as w i l l be seen i n Theorem 12. Why do we get more synthesis i n Theorem 10 ? Clues to this can be found i n the Notes section of §1.5 on page 31; given that we are using Meyer's theorem on harmonious sets of frequencies, the key to Theorem 10 i s Theorem 1 i n §1.5.1. On the other hand, with 'less Bochner', we lose the Salem elements in going from Theorem 10 to Theorem 11 (whereas i n the r e a l case, i t i s in the analogue of Theorem 10 i t s e l f , i . e . i n the weaker version of syn-thesis which Meyer obtains v i a harmonious sets, that the Salem numbers are lost) . Why do we get less of Bochner's property, so that Theorem 11 i s proved only for the Pisot elements? The proof of Theorem 12 i n §4 holds the ans-wer to that, and the reader must look there! In this proof, which looks very much l i k e that of the counterpart to Theorem 12 on the l i n e , there i s a key step where we need to know that the conjugates of 9 are a l l s t r i c t -ly less than 1 i n absolute value. On the l i n e , one needs only to know 107 that the conjugates of 9 are a l l less than or equal to 1 i n absolut value ( i n proving Bochner's property for C(9) when 9 i s a Pisot or Salem number). We w i l l do our best i n §4 to indicate just where th departure with the r e a l case takes place. ooOoo 108 §3. Outlines of the proofs of Theorems 9 and 10. Theorem 10 i s dealt with i n §3.2 on page 110. §3.1 Proof of Theorem 9 using Herz' technique. This exercise can be done with any set of the 'non-overlap' type; that i s , with sets described by (59) i n which Case I of §11.2.4 applies (see page 69). See sec. 3.2 of [7] for a discussion of Herz' c r i t e r i o n on the c i r c l e . We outline the steps i n the proof i n the simplest case. 1°. Let r = { z £ k{x l ] : L(z) < 0 } , A = A ( D , PM = PM(D For s i m p l i c i t y , choose 0 = x and I = { 0, 1 }, so that E = { Z E . x 1 : e. = 0 o t l l . 1-1 1 The dual of T i s G = k[x] ,-k 2 . For each k > 1 , 6 E i s a subset of the subgroup T k = { z : L(z) < - k } . -k Choose the sequence A^ E 0 E ( a r b i t r a r i l y ) . 3° . Choose any f £ A and u £ PM. We now obtain trigonometric polynomials f, that approximate k f i n norm; at the same time we obtain the required measures that approximate u weak*. 109 For each fixed k , l e t H be the f i n i t e subgroup of V : K. r k " i H. - ( X P. x : p. = 0, 1, .... or p-1 }, K i= 1 1 1 so that r i s a direct sum T = © • Now l e t f , be that function on T which has, for each h £ the constant value fCh + X,) on the coset h + of . Then f k "A - 11 F "A a n d " f " f k "A ° A S K "* °°* We deal now with U. For each basic open set I c T, y(I) i s another way of wr i t -ing < 1 , y > , the value of y on the trigonometric polynomial V For each k, l e t y, be the measure y, = I y(h+ r, ) 6. . , . k , „ k h+Ak h £ H k * Thus the 'pseudomass' di s t r i b u t e d on each coset h + Y i s K. gathered up and placed as a point mass at h + X . Then K. < f ' y k > = < f k , y > for each k, and hence 11 \ 11 PM - 11 y 11 PM a n d y k * y W 6 a k * • If f = 0 on E , then each f i s 0 on a neighbourhood of E and one can argue synthesis for E right away. 110 If y £ PM(E) , then y £ M(F ) , where for each k , k - i = ( I e. x : e. = 0 or 1 } + X. 1 = 1 This completes the proof. §3.2 Proof of Theorem 10 using Meyer's technique. The constant C and the f i n i t e set F c E = E(9,I) come straight from the fundamental theorem of Meyer on the atomizing properties of harmonious sets ( [28, p. 130, T h . X l ] ) . Harmonious sets of frequencies i n k{x }^ were discussed i n §§4 and 5 of Chapter I. Let G and T here both denote k{x Meyer works with bounded functions (on G ) . The spectrum of a bounded function (j> on G i s the support of the pseudomeasure y on T which i s the Fourier transform of cf>. Let us go quickly through the same steps as given i n the proof of Th. VII on p. 223 of [281 . We leave i t to the reader to take Fourier transforms to get Theorem 10. 1°. THEOREM (Meyer). Let A <= T be harmonious, and l e t E c T be compact. Then there exist a f i n i t e subset F of E , a con-stant C > 0 , and a l i n e a r mapping L such that : (a) i f cf> i s a bounded continuous function on G whose spec-trum l i e s i n A* + E , where A* cz A , then L(f> i s an almost periodic function whose spectrum l i e s i n A* + F; (b) sup I L<}) I G < C sup I 4> I ; G (c) I L<J)(z) - <J>(z) I < C 1 z I sup 1*1, G z £ G. Ill Remember we are working with E = E(9,I), where 9 belongs to T. Let A = A(8,I) be the set of a l l f i n i t e sums I p . 0 1 , i>0 1 where p, E I for every I . More s p e c i f i c a l l y , l e t k-1 A = U A ; A = { X p . 9 1 :p . e i } , k > 1. fc k i=0 1 1 By Theorem 1 i n §1.5.1 (page 26), A i s harmonious. Let C , F and L be the materials supplied by 1° for A and E. For each integer q , l e t T be the isometry of L (G) q q co T (j) (z) = (j)(9 z) , z e G, <j> £ L (G) . q For each k > 1 , l e t L be the lin e a r map defined by L k = T_ k o L o x k Start with a bounded continuous function cj) on G whose spectrum l i e s i n E. k The spectrum of lies i n 9 E = A k + E ; hence the spectrum of L o Tk(<j)) l i e s i n A k + F (using (a) i n 1°). It follows that the spectrum of L 4 i s contained i n k k k 3 ( A + F) = {X P.Q x : p . e I } + 6 K F . K i=l 1 1 112 From (b) and (c) of Meyer's theorem, we deduce that the trigonometric polynomials cp = L (cp), k >_ 1 , s a t i s f y and sup I cp C sup | cp | G cp k(z) - cp(z) C 19 ,-k IzI sup G This completes the proof of Theorem 10. 113 §4. Bochner's property when 8 i s a Pisot element, and the proof of Theorem 11. The purpose of this section i s to prove Theorem 12 below, from which Theorem 11 follows. Theorem 12 expresses Bochner's property for the r e s t r i c t i o n algebra A(E) , with E as i n Theorem 11. (Bochner's property for a Banach algebra i s discussed b r i e f l y i n [28, pp. 226-228].) The o v e r a l l structure of the proof i s i d e n t i c a l with that of [28, Th. V I I I , p. 228], with which Theorem 12 should be compared. The reader i s cautioned that the symbols G, F, r ... w i l l take on meanings i n this section d i f f e r e n t from those assigned to them i n e a r l i e r sections. C(X) denotes the continuous functions on the space X. §4.1 Bochner's property for A(E), E as i n Theorem 11, THEOREM 12. Let 8 be a Pisot element i n k{x - 1}, and suppose I i s a f i n i t e subset of k[x] which includes {0} U B , where B i s some basis for WT ... = {ui £ k[x] : I oo 1 < 1 0 1 ) . (This would hold, for example, L(0;-1 i f I => (0, 1, x, ..., x L ( 9 ) - 1 }•) Let E = E(8,I) be the compact set 0 0 - i of a l l sums X , e. 8 , where e. € I for every i . For each k > I, i=l i i — r k - i l e t F, be the f i n i t e subset of E given by F, = IX e.0 : e . € l } . k k i=i I i Then Bochner's property holds for E and the sets F : f € C(E) and sup II f II . v < °° implies k>l M V f 6 A(E) and lim II f || = || f I I . , . , . . k-*-°° M k ; M ; Choosing 8 = x and I = {0, 1, p-l} (our scalar f i e l d k) yi e l d s ; o Bochner's property for the group G = {z e k{x 1}: L(z) < 0} with respect 114 to the f i n i t e subgroups H = {Z a . x 1 : a. = 0, 1, or p - l } . 1=1 Theorem 11 now follows from Theorem 12 upon examining Bochner's prop-erty i n i t s dual form; see [28, p. 227]. We know already from Theorem 10 that the set E i n Theorems 11 and 12 obeys synthesis, hence PM(E) i s the Banach space dual of A(E). As i n [28, p. 227, Prop. 9], we obtain for each u £ PM(E) a sequence of measures , k :> 1 , such that £ M(F^) for each k, II y k I I p M < II y II p M , and y k -»- y weak*. Several steps w i l l be required i n the proof of Theorem 12, beginning with the d e f i n i t i o n of a compact metrizable group G which r e f l e c t s the algebraic properties of 9 , and of which k{x }^ may be regarded as a dense subgroup. §4.2 A compact group G (the proof of Theorem 12 begins) Let the minimum polynomial of 9 over k(x) be P(X) = X n + a,X n _ 1 + ... + a , 1 n where a_^  £ k[x] for each i . Since 9 i s a Pisot element, i t s conjug-ates 9^, 9^ s a t i s f y 19 J < 1 for 2 _< i <^  n, and the expressions for the c o e f f i c i e n t s a^, .. ., a^ i n terms of 0 and i t s conjugates show that (86) l a j = 191 , and la I < 181 for 2 < i < n. For each j >. 1. l e t r be the subgroup of k{x }^ defined by r . = 9" jk[x] + ... + 9 " j _ n + 1 k[x] . 115 Since 0 ^ + a, 0 1 + ... + a 0 -1 ° = 0 for every j , we 1 n see that V <= T. , , i = 1,2,... , and we define V to be the group J 3+1 which i s the union of a l l of the T.: V = U V. . Then T i s a count-3 3 able dense subgroup of k{x }. We consider T with the discrete topology, and l e t G be i t s compact dual group. Since V i s countable, G i s metrizable. Let h : k{x - 1} •+ G be the mapping which i s dual to the canonical i n c l u s i o n i : V -* k{x L}. Then h i s an embedding of k{x }^ onto a dense subgroup of G . Proposition 14 below shows that G can be regarded as a closed sub-co CO _ group of the compact group GQ - II GQ , the d i r e c t product of countably j = 1 -1 many copies of the group of integers GQ of k{x }. Recall that (z) denotes the f r a c t i o n a l part of any z € k{x l}. If (z) = 0, or what i s the same thing, i f z € k[x], then we also write z E 0 (mod 1). PROPOSITION 14. The compact group G . can be regarded as the group oo of a l l sequences x = (<p^ , cp^, ... ) i n GQ for which the congruence <p.+a.cf>.i1 + ...+a<p., = 0 (mod 1) 3 1 J+l n Tj+n holds for a l l j >^  1 ; the mapping h : k{x l} -»• G i s then given by h(u) = ( ( u 0 _ 1 ) , (u0" 2) , ... ) , u £ H x " 1 } . Proof of Proposition 14. F i r s t we describe a continuous i n j e c t i v e CO homomorphism of G into the indicated subgroup of GQ ; then we show the mapping i s also surjective, which makes i t a homeomorphism. 116 For each character X on T , l e t <}> , for each j > 1, denote the unique element of G Q for which Yj (*r<}> J = X(x r8 3 ) holds for every r = 0, 1,2, .... Then x( b 9 2) = Y ^ C 0 ^ ) holds for every b E k[x] and j >^  1. For r > 0 and j > 1, we have x r e - j + a V e " ^ 1 + ... + a x r e " j - n = 0 from which we obtain (87) Y x r (*j + a ^ . + 1 + ... + a ^ ) = 1. For each j >. 1> (87) holds for a l l r > 0, and this implies (88) <j>. + a.*.., + ... + a (J).. = 0 (mod 1), j > 1. j 1 J+i n j+n — Conversely, l e t <j)^ , cb^, ... be a sequence i n s a t i s f y i n g (88). For each j > 1, we can define the character Xj o n F\ by setting j+n-1 _. j+n-1 (89) X-( Z b ^ 1 ) = Y X ( X b i<j> i), b. £ k[x]. 2 i = j 1 i = j 1 1 1 Since { 8 3 , ... , 8 2 } i s a basis for the free k[x]-module r , Xj i s well defined by (89). Moreover, i t i s easi l y seen from the i - J - 1 - ... - a e- j- n , n congruences (88) and the fact that 8 3 = -a^ that X j + ^ I = Xj f ° r every j :> 1; hence we can define the char- -acter )( on T by setting X = Xj o n each r . Clearly, i f x i - s t n e r e s t r i c t i o n to T of the continuous character Y u , then x( b 8~J) = Y 1 ( b u 9 ~ J ) holds for a l l b £ k[x] , from which we see that the corresponding sequence i n G^ i s given by <l>j = ( u 8 " j ) , j > 1. 117 § A.3 Extension to G of Fourier transforms of measures supported i n F If y £ M(F ) for some k >_ 1, then y i s a f i n i t e trigonometric IS. sum whose frequencies belong to -F (which i s contained i n V) ', hence there i s a unique trigonometric polynomial P on G , whose frequencies and Fourier c o e f f i c i e n t s are those of y , such that y = P o h ( h i s our embedding of k{x } into G ) . This follows because h(k{x _ 1}) i s dense i n G , and for the same reason we have (90) II y II = sup l y l = sup IP I, y = P o h . k l x " 1 ) Let E be the vector space of a l l such P on G , with norm as given by ( 9 0 ) . (§4.4, with the conclusion of the proof of Theorem 12, may be read at this point, but the lemmas below w i l l be needed i n the f i n a l e . ) We now introduce a class of measures which w i l l play a role i n d i s t -inguishing G^ = h(k{x l}) from i t s complement G^ i n G. For each j >^  1, l e t V^ . be the measure which di s t r i b u t e s a unit mass evenly among the L(8) +1 points of {0} U { £ 9 ~ j : e € B } . Here, B i s the basis for w L ( 0 ) _ i _ which i s assumed, i n Theorem 12, to be contained i n I. For positive integers p and q , the measure y i s p ,q defined as the convolution product p,q p+1 p+q Evidently, y i s carried by 9 P F for each p and q, p,q q 118 LEMMA 6. For each continuous function f on E, lim [ f du = f(0) uniformly i n q > 1. p + o o E P ' Q Proof of Lemma 6. The support of y l i e s i n 6 P E , which i s P 5 q contained i n an a r b i t r a r i l y small neighbourhood of 0 provided only p i s large enough. Now the transforms of the measures y are given by p.q y (u) = (L(6) + p.q _ pt-q / . \ 1 ) q n 1 + I Y_ ( e9~ J) ), j=p+l ^ e£B u J from which we see that the continuous extension M of y to G p.q p.q i s given by -a V^ I \ ( 9 1 ) M (x) = ( L ( 8 ) + l ) q n 1 + I Yc.(-<|).) • p,q - <r , J p+q n i J - T Y V - 1 | ) ) j. j =p+l V ££B 2 J LEMMA 7. For each fixed p > 1 and each x. e G2 = G ^  h(k{x 1 } ) , we have lim M (x) = 0. q -v oo p»q ~ Proof of Lemma 7. Evidently there i s a constant c < 1 such that each factor n. = ( L ( 9 ) + l ) _ 1 ( 1 + I Y ( -<J>.) J X e£B E 3 in (91), i f not 1, s a t i s f i e s I n. | < c. Suppose now that M ( x ) J — p.q ~ does not tend to 0 as q •> 0 0 (for some fixed p ) . Then there must exist a j . > 1 such that J0 — n j = 1 ( 3 ^ j 0 } ' 119 and this implies that y (-<p.) = 1 for a l l £ £ B and j > j . Thus e 3 o i f i > i , cb i s annihilated by each member of WT , n N , , and hence J — J o T j L ( 8 ) - l (92) L(cpj) < - L ( 8 ) i f j > j Q . It follows from (92) then that we must have (93) <J> + a 14> j + 1 + ... + a n<p j + n = 0 for a l l j > j Q , because the expression on the l e f t here, already = 0 (mod 1) by the d e f i n i t i o n of G, can have no i n t e g r a l part (the c o e f f i c i e n t s i n (93) sa t i s f y l a I < 181 for i = l , . . . , n ; see (86) i n §4.2). Now the lin e a r recurrence r e l a t i o n (93) can be solved i n terms of 8 and i t s conjugates: there must be elements u £ k{x l} and u 2' , u £ K (the algebraic closure of k{x l}) for which (94) (j> = u 8 J + u 2e 2" J + ... + u n e n j , j > j Q . But 8 i s a Pisot element: |8„| < 1, |8 I < 1, and we can l n conclude from (94) (cf. Remarks and Lemma 8 below) that i n fact +j = for j > j Q . Since for a l l j > 1 we have <p.+ a.cp + ... 4- ad> = 0 (mod 1) J 1 J+l n j+n and 8" j 4- a ^ - J " 1 4- ... 4- a 0- j" n = 1 n we obtain, by a decreasing induction on j , that <t>. = ( u 8 3 ) , j > 1. 120 This means that x = (cp^, cp^ , ... ) belongs to h(k{x }, and the proof of the c r u c i a l Lemma 7 i s complete. REMARKS. We needed to use Lemma 8 below i n the proof. Either of parts (a) and (b) can be used to proceed beyond (94) i n the proof of Lemma 7; part (b) can be used because i t follows from (93) and (86) that <p. •+ 0 as j 0 0 (but remember that 0 needs to be a Pisot element for (86) to hold). We note that (94) i t s e l f could have been reached with a Salem element 0. Now while only one part of Lemma 8 i s actually needed for Lemma 7, our proof of Lemma 8 proceeds by induction on n along both partp. LEMMA 8. Let u., . . . , u and v., . . . , v be 2n elements of 1 n 1 n the algebraic closure K of k{x *}, with the v's a l l being d i s t i n c t . Let ( f ( j ) } . , be the sequence i n K defined by (a) Assume that f ( j ) i s bounded for j >. 1• Then for each i for which we have |v.I > 1, u. must be 0. l l (b) I f i n fact f ( j ) -> 0 as j -»• 0 0, then u^ = 0 for each i such that Iv.I > 1. Proof of Lemma 8. Clearly both parts of the lemma hold when n = 1. Assume then that they hold for n-1 , and reorder the v's so that J j > I-= Iv I > I v m m+1 I > > I v | . n 121 If I v^ l < 1 t h e r e i s n o t h i n g t o p r o v e , so suppose Iv^ l >^  1 ( w i t h |v^| > 1 i f ( f ( j ) } i s merely bounded). For i = 1, ... , n , l e t p. = v. v ~ 1 . Then I p. I = 1 for i < m, x 1 1 1 — and lp J < 1 for i > m. Since f ( j ) v ^ 3 tends to 0 as j -»• °°, so also does g(j) = u + u p J + ... + u p J . 1 Z L mm' The proof w i l l be completed (by induction) i f we can show u^ = 0. Now this must be so i n the case that m = 1, so suppose m > 1 and consider g ( j + D - g(j) = u ( p - l ) p j + ... + u ( p - l ) p j . Z Z Z m m m This tends to 0 also, and since 1, p „ , . . . , p are a l l d i s t i n c t , Z m our inductive hypothesis implies that u„ = ... = u =0. Z m Hence = 0 as well, and the proof of Lemma 8 i s complete. §4.4 Enter the Riesz representation theorem (the proof of Theorem 12 concludes) This i s now a straight t r a n s c r i p t i o n of the conclusion for the r e a l case (cf. [28, p. 231]) . Let C = lim II f || ._ . , and l e t L be the linear functional k - H » k defined on E as follows. If p E M(F ) and P oh = y, then K. (95) L(P) = / f d u F k (see (90) and the opening paragraph i n §4.3). P l a i n l y one has L ( P ) I < II f II sup l y l < C sup IP | 122 By the Hahn-Banach theorem, L can be extended to a l l of C(G) with no increase i n norm, and by the Riesz representation theorem there i s a measure a, II 0" II < C , such that L(P) = / Pda. Thus (95) becomes G / f dy = J P da for every y supported i n one of the sets F , where P i s determined from y according to Poh = y. Now a can be written as 0^ + a^, where i s the r e s t r i c t i o n of a to G^ = h ( k { x - 1 } ) , and i s the r e s t r i c t i o n to G^ = G ^G^,. With the help of Lemmas 6 and 7, one sees that = 0. But then o~ can be written as h(x), where T i s a Borel measure on klx" 1} of norm || x II = II a II < C. We obtain / f dy = / y dx = / x dy k{x - 1} k{x - 1} k{x - 1} for a l l y 6 U M ( F , ) • This implies that x = f on each of the sets F ; hence T = f on E, since UF i s dense in E. K. rC Thus we have f e B(E) and II f I L < C. (See i 28, p. 115], for example, for the d e f i n i t i o n of the r e s t r i c t i o n algebra B(E).) But A(E) and B(E) are the same thing when E i s compact; they are isometric (e.g. [28, p. 121]), and Theorem 12 i s proved. 123 BIBLIOGRAPHY 1. L. A. Balasov and A. I. 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