Construction techniques for some thin sets in duals of compact abelian groups

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D. J. H ^AJELA

Construction techniques for some thin sets in duals of compact abelian groups

Annales de l’institut Fourier, tome 36, n^o3 (1986), p. 137-166

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Ann. hist. Fourier, Grenoble 36, 3 (1986), 137-166

CONSTRUCTION TECHNIQUES FOR SOME THIN SETS IN DUALS OF COMPACT

ABELIAN GROUPS

by D.J. HAJELA

1. Introduction.

In this note we give construction techniques for A(p) sets in various groups. As a consequence we are able to recapture previous results on these sets (see e.g. [2], [4], [6], [23]) as well as prove some new results. In particular we show that in the dual of any compact abelian group there exists a A(4) set which is not A (4 + e) for any e > 0.

We now describe the contents of this note more fully. Let us recall the definition of a A(p) set for G a compact abelian group : If F = {7,^=1 C G* (G* is the dual group of G) then F is a A(p) set if there exists a constant Ap ^ > 0 such that,

n

J. '''"

^{^(^ ^}^A^{?^} ^{/ = = 1}^{S ^;} ^V⁰⁾ ^(1.1)

for some 0 < q < p , for all n e N and all (a,)^ E C " . By an application of Holder's inequality it is easily seen that if the above holds for some 0 < q <p then it holds for all 0 < r < p (see

[23]).

In section 2 we show that in the dual of the Cantor group D*

(where D == {— l , ! } * ^ ) there exist A(p) sets which are not A(p -+- e) for any e > 0, where p = 2k and 2 < k E N . Some of the results in this section follow from more general results in the following sections, but we have given proofs specifically adapted to D . This is because the construction in D is particularly

Keyword: A(p) sets.

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revealing and shows some of the basic ideas used in other constructions.

Essential to the construction for D are some ideas from coding theory and in particular it was some remarks of Johnson, Schectman, and Wilson (unpublished) in the p = 4 case that led us to general case for D and subsequently to other groups.

In section 3 some preliminary results used for the rest of the paper are proved. In particular the study of A(p) sets for general compact abelian groups is reduced to the study of A(p) sets in a few special groups, namely in the dual groups Z , Z ( p ^ ) ^ Z(^) ^ . . . (for an increasing sequence of primes (?„)), Z(p00) = U Z(p")

n>Q

(p a prime) and Z (p) 0 Z(p) ^ Z(p) 0 . . . . This is effected by using the results of [6], where this type of idea was used in showing that there are sets which are A(p) for all 1 < p < °° but which are not Sidon sets.

In section 4 we give construction for Z and Z ( p ^ ) ^ Z(p^) 0 . . . While constructions for these two groups areknown(see [23], [6]) we give a construction based on a theorem of Bose and Chowla [5]

(which was used to assert the existence of finite projective planes).

In this section we also generalize a method of Erdos (see [7]) which shows that with respect to a certain biased coin tossing measure on the space of integer sequences almost all sequences have a prescribed rate of growth and that an arbitrary integer can be written in a bounded number of ways as a sum of elements of a given random integer sequence. This result easily yields that for p = 2k, 2 < k E N almost all integer sequences are A(p) but not A(p + e). We also give in this section a more precise version of the growth of the A (4) constant of the squares than in [23]. It is somewhat surprising that the sequence constructed in the p = 4 case above are like squares, since the squares are not A(4).

In section 5 we turn to the dual group Z(p) ^ Z(p) 9 . . . for p > 2. It is shown that for p > k > 2 there are A(2k) sets which are not A(2k -+- e) for any e > 0 by using certain classical facts about symmetric polynomials. For k > p we don't have a construction but it is shown that one possible approach is to reduce the problem to one about counting rational points in a certain variety. This problem in algebraic geometry however appears to be rather delicate. The results of section 2 along with those in section 5

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CONSTRUCTION OF A SETS 139

constitute a strong form of a solution to a problem in Lopez and Ross's book "Sidon Sets", page 171 (see [19]).

In section 6 we turn to construction in Z(p°°) = U Z i p " ) . w > o

It is shown that there are A (4) sets which are not A (4 + e) for all 6 > 0 . One also obtains construction for A(2k) sets (k>2).

The main idea here is that one may reduce to a well known theorem ofTuran's in extremal graph theory.

In connection with the above results, one should mention a result of Pisier (unpublished). His result is: Given A C G^, I A | = n and given 6 > 0 3 B C A , |B | > n^2-8 with \(B) < C^ (where Cg doesn't depend on n and ^(B) is the A(4) constant of B).

The interest in such a statement is that the A (4 + e) constant of B should be large by suitably choosing 5(e). One should therefore be able to glue such B's together to find bad A (4) sets. The problem of course is that one doesn't obtain A(4) sets which are A(4 + e ) for all e > 0. Also the proof is limited to A(4) sets. Let us finally point out that the gluing process could be non-trivial as will be seen in section 6.

We will use standard notations and any notation not mentioned in the paper may be found in [22], [27] and [18]. Let us just mention that |S| denotes the cardinality of a set S and [x] denotes the greatest integer function for x E R .

We wish to thank P. Deligne, J. Foumier, W. Johnson, G.

Pisier, D. Ray-Chaudhuri, K. Ross and G. Schectman for useful comments and communications.

2. Construction in the dual of the Cantor group.

In this section we construct a A (2k) set in D* which is not A(2k - h e ) , for all e > 0, 2 < k G N , where D = {- 1 , l}^ is the Cantor group. Recall that the set of characters for D is the Fourier —Walsh system : For x = (;c^) E D we let e^ : D —> { — 1,1}

be defined by e^(x) = x^ where fcE N , k > 1 . Then an element of the dual group D* of D consists of finite products of the e^.

Given a finite subset A of N let us write, W^ = n e . /eA

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2.7 Preliminaries.

To construct A(p) sets which are not A(p + e) we shall use the proposition below which states that among certain tuples of O's and I ' s one can find a large set of tuples whose elements when added together have distinct sums.

PROPOSITION 2.1.1. -Let 2 < w € N and let n G N such that n >[log2 m + 1] + 1 . Then among the 2^ tuples of O's and I ' s ie. {0,1}^ one can find a subset A C {0,1}^ vw'rA the following properties:

1) | A | == 2"

2) z^r fc e N , l < k < w W /6?r { c ^ , . . . , c^} c A ^ri {^+1. • • • ^2k}^{c A w}^ { c i , . . . , c^} n {c^,,.., c^} = 0 . Then c^ + . . . 4- c^ ^ c^ + . . . + c^ (the addition of two tuples is performed coordinatewise modulo 2).

Proof - Let GF(2") denote the Galois field of 2" elements where n is chosen so that n > [log^ m + 1] 4- 1 and m is fixed.

Regarding GF(2") as a vector space over GF(2) we have that dim GF(2") = n. So let [ x ^ , . . . , x^} C GF(2") be a basis for GF(2") over GF(2). To xeGF(2n) we associate the n-tuple of O's and

n

\'s(a\,. . . , a\) where ^ = L a\ x ^ . In what follows we will always 1=1

maintain the order of the x / s in any expansion of a given element of GF(2"). In a similar fashion associate to each odd power

n

x^-1 of x its ^-tuple ( a ^ , . . . , ^ ) i.e. x^'1 = S flf ;c, for i= i

1 < k < m. Finally associate to x the mn tuple of 0'5 and 1 ' s a(x)= ( a ; , . . . , ^ , . . . , ^ , . . . , ^ , . . , , 0 7 , . . . , ^ ) . We claim that A = {a(x) |x eGF(2M)} has the desired properties. Clearly I A | = 2" because the first n coordinates of a(x) are the basis expansion for x . To see the second property let { y ^ , . . . , y ^ } C A and [Yk+i, • • • , y^c} c A where the two sets are disjoint (note that the condition n > [log^ m -»- 1] -t- 1 assures us that such sets do exist for all \<k<m). Pick z/GGF(2") so that a ( z , ) = j / , ,

1 < i < 2k. Now suppose that:

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CONSTRUCTION OF Ap SETS 141

k 2k S Yi= S y , . 1=1 i = f c + l

(2.1.1)

Then by virtue of a(z,) = 3^ we have that:

2Z^ -r . . . -r Z^ - Z^.n -1- . . . -h Z^27"1 + + 727"1 = 72 / - l 4- -4- 72f~l (2.1.2) for 1 < / < f c . Now since GF(2") has characteristic 2 the above condition forces:

y/ -(- -L y/ = -/ -L J. -7

Z . ^- . . . -T £, . ^I.-L, ^ . . . • Z^ for 1 <f <2k- 1. (2.1.3) This follows by taking 1 < / < 2k — 1, writing it as / = 27 / ' where /' is odd and raising equation (2.1.2) corresponding to / ' to the

^th power. Letting M, = (1 ,z,,. . . , zf~1) for 1 < i < 2k the last equation in turn forces {u^^ to be linearly dependent. So,

^-1

det ^Z²⁷^{2 f c}^~^l

z; • • • zf = 0.

^Tk ^ " ' -2^-1

z^

But the above is the Van der Monde determinant and is so also n (z, — z.) ^ 0 since the z, are distinct. This contradiction means

i>J

that A has the second property, n Remarks. —2.1.1) Notice that the above type of result is the best possible of its kind in the sense that if one is given a set S with a binary operation + , which has the closure property with respect to S, then for the maximal subset A C S with the second property in the above proposition one has Bin | A | / | S j^ < 1 (as

| s | —> oo). in the proposition the set A has ( A | = T = | {0,1}^ j^ .

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2.1.2) In view of the remark above it follows that the result in the proposition doesn't generalize to the characteristic p case (p > 2) with the above proof. To see this fix 1 < m E N and choose a prime p > 2m — 1. In order for the above proof to work and to choose a maximal subset (as in the above remark) one must choose exponents k, , . . . , k^_, E N , 2 < k, <. . . < k^_, s.t. the

\ 2fc

conditions i x^' == ^ x^i f == i , . . . , ^ - i , k fixed,

i = l i=k+l ?

k < m and the condition ^ + . . . 4- ^ = x^i + . . . 4- x^ for x,EGF(7/1) force the conditions

k 2k

I x/ = S x/ 7 = = 1 , . . . , 2fc- 1.

1=1 j = f c + i

If k = m for example this is clearly not automatic (since p > 2m — 1). In some sense what is special about the p = 2 case is that any 7 < 2w — 1 can be written as 7 = 2^\ with 7\ = 1 (mod 2) and all the odd exponents have already been chosen

so that the conditions are forced.

2.1.3) For fields whose characteristic is not 2 an alternative approach is discussed in section 5. As an example of one case in the non characteristic 2 situation, pick a prime p > 2 . Set

A = { G C . ^ I J C G G F O ^ ) }

(here we expand x and x2 in tuple fashion regarding GF(p") as a vector space over GF(p)). Then | A | = p" while

| { 0 , l , . . . , p - l}2"!^3"

and if a , b , c , d are A (all distinct) then a + b ^ c 4- d . To see this simply observe that x + y = w -t- z and ;c2 + >/2 = w2 -h z2

have no solution with {x , y ,w ,z} CGF(pn) and with x , y , w , z being distinct.

2.1.4) With m = 2 in proposition 2.1.1 the proof shows that A = {(x ,x3) | x G GF(2")} works. This case corresponds to a standard construction in coding theory (see remark 2.1.5).

2.1.5) We finally point out that the construction in the proof of the proposition is the same type of construction as that of certain well known cyclic codes (particularly BCH codes) (see [26]). This resemblance was pointed out to us by D.K. Ray - Chaudhuri.

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2.2 The Construction..

To begin with we set up a correspondance between certain subsets of the Walsh functions and the sets constructed in the proposition.

Fix 2 < m € N and choose any n > [log^w + 1] + 1 . Let W^ be the Walsh functions generated by e ^ , . . . , e^. For W^ ^ W^

construct the following tuple of O's and I ' s (of length mn) : , - ( 1 if ie A

sw*(')= J O i f , < ; A ,

Note that given W^WpGW^, and their associated tuples S^

and S^B the tuple for W^ W^ ,S^WB = SWA + ^B (Mod 2 )- By the construction in proposition 2.1.1 it follows that we can find a set A^ of Walsh functions satisfying the following properties :

1) A ^ C W ^ 2) I A ^ I = 2 "

3) If WA^ A^ , ? = 1, . . . , 2k are distinct Walsh functions

k 2k

then n WA.^= n WA. provided that A: < m .

,=1 ! i=k+l l

Now pick ^ so that n^ = min {27 | 27 > [log^ m + 1] + 1}.

Now define ^4.1 = 2^. for / > 1 ,7 G N . Finally put E = U A^..

/ > i /

Note that A ^ ^ . C A ^ ^ . ^ ^ , by the construction and because GF(2"/+1) DGF(2n/) as a subfield because ^l^.+i . We will show that :

1) E is a A(2m) set

2) E is not a A(2m 4- e) set for all e > 0.

We show (2) first. This easily follows from some material in section 3. We choose to give however a different proof than is usual by using some well-known techniques from the local theory of Banach spaces. Fix e > 0 and put p = 2m + e. Then [W^l (closed linear span of W^. in p-norm) is isometric to /2W/W , for a fixed / > 1. This is because of the obvious fact that if D^. is the set of dyadic intervals of length 2-w/w on [0,1] and I ^ D ^ . then Xi E V^mn^p (of course here we are looking upon the e^ ' s

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as Rademacher functions on [0,1]). Let Xp(A^.) be the constant of equivalence between the Lp and L^ norms on

^mn^p' If E is a h(pY set then Xp(A^.)<A where A is the A(p) constant of E. Now dim [A^.]p = 2"7. It is a well known fact that the maximal dimension of Hilbertian subspaces which are uniformly enbeddable in ^nfm is less than 2"7. It follows

\ ^wn? —> + °° as / —> + oo . This is a contradiction. We give a more precise computation of Xp(A^.) below. The proof here is easily adapted from [10].

PROPOSITION 2.2.1. -For any m > 1, / > 1 and e > 0 we have that Xp(A^.) > c(p) \ A^. ^-^P ^here p = 2m + e and c(p) = l/p17^ .

Proo/ - Let v^, . . . , H^ be the Walsh functions in

^mn^ = 2 7). For any choice of scalars (0?^ ^ we have,

(2i'.,'i) «,' ir»

^fc

/ = 1

S ^.w, <X,(A^)(^ |a,|2)1/2 . (2.2.1) Let 0,)^=i be the Rademacher functions and let (x^^C /2W7W

be vectors which correspond to Wj under the isometry between [W^]p and ^m . Let x, = (x,^,)^, where n = 2"^ . By the left hand side of (2.2.1) and Khinchin's inequality we have,

r

1 k

ll

p

k^ <J, ^ .,(r)x, rfr

fc

n /l! V P

= S J,^L ^^',i dt i= i ° /= i

n k

<B,£(2l.,,|').»

1=1 / = 1 (2.2.2) where Bp is the upper Khinchin constant. By dualizing the right hand side of (2.2.1) we get that,

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k

(Sl^J^/^X^A^.). (2.2.3)

By plugging (2.2.3) into (2.2.2) and using that Bp < p172 (for p > 2) it follows that,

^ < B^ X^(A^) = B,^ X^(A^.) < p V 2 ^ ^(A^.).

It follows that: \ (A^.) > c(p) | A^. l^-m/p ^ We show next that X^ (A^,^.) < c where c = c(.m). It follows immediately that X^ (E) < c , since A^. C A^^.^ .

PROPOSITION 2.2.2. - For any w > 2 and ]> 1 w^ Aav^

^-2^ (^n? ^ c w^^^ c = c(m).

Proof. - Fix 7 > 1. Let w ^ , . . . , WN be the elements of A^.(N = 2^'). Set / = ^ fl,w, where (a,)^ E CN and

N

set A = ^, | a, |2 and B = ^ a,a.w,w.. We first observe that, 1=1 i^/

\fBk\<c,Ak (2.2.4) where c^ depends only on k and w (here 1 < k < m). This is because:

/B- =f(^a.-a^^)k

= f S a.., 5,,' . . . a^ a^ w, . . . w , w^. ...w,.

'1^1'...., ^'jl

By the second property in proposition 2.1.1

./\...v^ w ^ . . . w , ^ = 0

unless there is a pairing so that (y = i,. or ;'„ for some / and n for each 1 < / < k and similarly for ?}' (1 < / < k). So

\f^\< S l^ll^l...|a;j|^|. (2.2.5) all pairings

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For a fixed pairing P we certainly have

Ak ^Sl^il | ^ v | - • • |%| |^'(

p

since A^ has all products of length k of squares i.e. |a,|2. Note that only a finite number of pairings exist and this number only depends on m (and k). So (2.2.4) follows.

N

^ 1=1 ' ^ = 1

||2m _ A W -_,a

Now |/|2 = ( ^ fl,w, ) ( ^ ^. ) = A + B .

v1 = 1 / \ = i /

It follows that I/II^^A^ and

11/1^ = /(A + B r = / f (^A-^^w

k^O ^

=A"' +fmAm-lB+f S (m) Am-f cBt

m fc^'2 v k

A" + I (m)^m-kf^.

k>1

So

— 7

| | ^ | | 2 w - | [ ( f | | 2 w | < A W 4 . y ^ I A'"-^ FR^

1 1 ^ ||2w "- I II/ bw | ^ A -h ^ I ; A J B i^ •'». ^ ' w '

A : > 2 W w

<A" 1 . £ (^{< W \}

<^

m)

f c > 2

with the last inequality following by the use of (2.2.4). Setting

= c^m)^ = 1 + 1 , ( m) Of, we have

^im ^ ^

k > 2 k

Ifi^^c^ 11/11^ and so ||/||^<c _{1 1 2 •}

Remarks. - (2.2.1) The result in proposition 2.2.2 easily follows from material in section 5, but the proof above is somewhat different from that in section 5 and is especially simple.

(2.2.2) The reader will observe that we could also have built our example on "disjoint" blocks A^n. instead of "inductive" ones

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(i.e. A^. C A^.^). We choose to use the latter type of blocks just because this feature was implicit in the construction of propo-

sition 2.1.1.

3. Preliminary facts.

In this section we state some simple results which are used in the rest of the paper. We start with a result which states that A ( p ) sets are thin from the point of view of the groups they contain. A generalization of this result is in [6] and the proof in [6] was based on ideas in [23]. From now on if E is A(p) set for p>2 then the A ( p ) constant of E, \p (E) is the constant of equivalence between the Lp and L^ norms on L^ = { / E Lp \f(\) = 0 if x ^ E}.

PROPOSITION 3.1. - Let G C F (dual of some compact abelian group) be a group wth | G | < + o o . Let A be A(q) for some q>2. Then \ (A)> |G H A I ^ / I G |l / q .

Remark 3.1. — It is obvious from [6] that the above result is valid not only for finite groups but also translates of finite groups.

The next result improves the estimate of the A ( p ) constant involved over that in [6]. It is an obvious modification of the proof in [6]. A similar estimate appears in [4] but the proof is somewhat different. We require the following definition.

DEFINITION 3.1. - Let r be an abelian group and let 2 < ^ E N . For all A C F denote by R(A,n) all functions

f: A — — > Ns.t. S f(x)-n.

X ^ A

For 7 G r, R (A, n, 7) denotes all fs.t. ^ X^^ = 7 and

X € E A

/ G R ( A , n).

PROPOSITION 3.2. — Let G be a compact abelian group "with dual group r and assume that \ R (A, n, 7) | < M for all 7 € r and some A C r . Then \^ (A) ^(M^!))172" (here n>2).

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Proof. — Let /= ^ flOc).^ for some finite set A C A. By the x e A

multinominal expansion we have

r= I ^! ,, n (a(x)x)^

^e:R(A,M) n ^(^)! ^ € A

j c ^ A

= ^ ^ 7 where ^ = S - "! „ n aW^.

rer ' <reR(A,^,-r) n g(x)\ x<=A

x S A

Now by Holder's inequality

l^l2^ S ( n "! ..^ n l^^)!2^^

^ V ^ e R C A ' n . ^ Vⁿ ^(^x)^! /^{X £ A}

jc€ A

/ M ! \

sup — — — - . , ) I R ( A , » , 7 ) 1

\ ^ e R ( A , M , 7 ) n ^(.y)!/

x xeA /

(

^{, \}

< M ^ ! ^ —————) n \a(x)\2^x\

^ G R ( A , M , 7 ) n g ( x ) \ l x ^ A

x e A /

so ||/||^=Z|^|2 (by ParsevaFs identity)

<M(^!) I: 1: (-^n-^ " I^MI2'^

7er ^RCA^^)^^^^)¹/^{X € E A}

(

^{f \}

< M ( ^ z ! ) S ——————-^ n \a(x)\1^

^ e R ( A . M ) n ^ ( X ) ! ^ € A

x G A /

= M ( ^ ! ) ( S | a ( ^ ) |2) " (by the multinomial expansion)

=M(n\)\\f\\^.

So X^^AXd^Ai!)¹⁷²". D

The last result of this section reduces the study of A ( p ) sets for general compact abelian groups to a few special cases. A result

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of this type was stated in [6] for the purpose of studying Sidon sets.

We start with the following obvious proposition (for a proof see [6]).

PROPOSITION 3.3. — Let G be a compact abelian group and H a closed subgroup. Then A is a A ( p ) set in (G/H)* (1 <p <oo) if and only if A is a A ( p ) set in G*.

Proposition 3.3 shows that we may reduce our study to A ( p ) sets in the list of groups in proposition 3.4. For a slightly different proof of this simple fact, see [6].

PROPOSITION 3.4. — Let G be an infinite abelian group. Then G contains a subgroup of one of the following types :

I ) Z 2) Z(pi) ^ Z(py) e • • • f01" some increasing sequence of primes (pn)^=i 3) Z(p°°) and 4) Z ( p ) ^ Z ( p ) ^ . . . for some p (prime).

Proof. - Let r(G) be the torsion subgroup of G. If r (G) ^ G then G D Z . So assuming r(G) = G write G == e) G where Gp are the p-primary components. If there are infinitely many components then G D Z(p^) ^ Z(p^) ®. . . . I f there are finitely many components then j Gp j = 4- oo for some p . Then G (for this value of p) contains a basic subgroup B (see [21]). Denote by a(B) = sup 0(fe) where 0(&) is the order of b. If

& € B

a(B) = + oo then B will contain infinitely many cyclic groups in it's decomposition, so B D Z(p) Q Z(p) 0 . . . . If a(B) < oo then Gp = B C Gp/B (see [21]). If Gp/B = {0} then Gp = B and so B will contain infinitely many cyclic groups in its decomposition and so B D Z(p) ^ Z(p) ^ . . . . If G^/B ^ {0}

then Gp/B = 2 0 Q 0 Z(p°°), because Gp/B is divisible. Since Q is not torsion and Gp is, G /B = 2 ^ Z(p°°). So at least one Z(p°°) appears since Gp/B ^ {0}. It follows that G D Z(p°°). n

4. Constructions in Z and Z(p^) ^ Z(p^) ^ . . .

Constructions in Z and Z(p^) ^ Z(p^) 0 . . . are well known ([23], [6]). We will give a slightly different type of construction here.

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Our starting point is the following theorem of Bose and Chowla (see [5]).

PROPOSITION 4.1. —Let m = p" (where p is a prime, n G N^

and q = (m^1 — \)/m — 1 for some r G N . 77i^ we can find m + 1 integers (less than q) d^ = 0, ^i = 1, d ^ , . . . , d^ s.t. the sums df -+- . . . 4- df , 0 < ^ < ^ < . . . < ; , . < w ar^ all different mod ^.

Proposition 4.1 is for m being powers of primes (and this was what was needed to construct finite projective planes). Proposition 4.2 is an extension. A similar argument appears in [6] with a different conclusion.

PROPOSITION 4.2. -If n> 3m6m (for some m G N) then we

^Im

can find A C Z^ such that \ A | > ^—„„- and the sums (3 m)

a, 4- . . . + a, flr^ distinct mod ^ w/^r^ {a. , . . . , fl, } C A and

i < ^ < . . . 2 ^ .

^1/m

Proo/ — Choose n as above. Set x = (iiyn\+i—r/,,," • Then x > 2 and so there exists a prime p (by Bertrands's theorem, [15]) s.t. [x] + 1 < p < 2 [;c] + 2. So ;c < p < 2x + 2 < 3x (since

^l/w ^l/w

x > 2) i.e. there exists a prime p s.t. ^ ^ ^ < P <~^rn~^ •

ym+l-l

Set ^ = ————. By proposition 4.1 there exists a ^ , . . . ,'flp (less P ~" !

than q^) s.t. m-sums of the a's are distinct (mod q). Set A = { f l i , . . . , Op} . Then we have that

a. + . . . + a^ < mq < 3pm m <n.

So the m-sums are also distinct mod n. Also

^l/m ^\lm 1 ^ 1 = P > g l / w + l ^ l / m "~ / ^ w + l ^ ^ l / w •

Since it is not particularly important as to how large n should be in proposition 4.2 to make it true we could have used the prime

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number theorem instead of Bertrand's theorem in the proof above.

This is because for all e > 0, TT((I + e)n) — ir(n) —> + °° as n —^ + °° (and so there is a prime p , n < p < ( l - ^ - e ) n if n is large) by the prime number theorem. This would have yielded a somewhat larger set A in proposition 4.2. Since this is of no importance in what follows we choose to use Bertrand's theorem.

One may now easily construct A(2k) sets in Z which are not A(2k - h e ) , for all e > 0 .

PROPOSITION 4.3. — There is a set F C Z which is A(2k) but not A(2k 4- e) for e > 0, where k > 2 .

Proof. - Let (?„) be an increasing sequence of primes. By proposition 4.1 there exist sets E C Z s.t. \E \>p

1

^{_fc+i ,'•}

e,co,^'

£„ C 0,—"———— s.t. fe-sums out of £„ are distinct (we are now adding in Z and looking upon these sums). Set F^ = E^ and set a^ = k max F^ (for ( > 2) and F{ = a^. Set F = U F^.

i>\

It is clear that F is A(2k) by proposition 3.2. To see F is not

^+1 - 1

A(2k + 6), let A^ == ^ m \ m E N , 0 < m <—1———— and note Pn ~ 1

that | F^ n A^ | > ^ > ^ ^ 1 A^ [1/* . By a theorem of Rudin the cardinality of the intersection of a A(2A: + e) set with an arithmetic progression can't be so large (see [23], one can't quite use proposition 3.1, but certainly one can use appropriate generalizations of it. Since this is the only time we need anything other than proposition 3.1 we don't state the general results), a It should be clear that by using proposition 4.2 on "disjoint blocks" of Z ( p ^ ) ^ Z ( p ^ ) ^ . . . , one may build analogous examples.

PROPOSITION 4.4. - There is a set E C Z ( p ^ ) ^ Z(p^) ^ .. . which is A(2k) but not A(2k 4- e).

Proof — Assume without loss of generality that p^ > 3k6k

for all n > 1. By proposition 4.2. there exists E^ C Z(p^) s.t.

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i/k

I ^n I s> ^n— i ^ and fe-sums out of E^ are distinct mod ?„.

(3 k )

Set E == U £„ with each E^ embedded in Z(^) e Z(p^) . . .

n= 1

in the cannonical fashion. By proposition 3.2 E is A(2fe) and since

X^(E)>|EnZ(p,)|^/^/ u + e

> ————.7. P^~l/2fc+e -— + oo as n -^ + oo

O^fc)^ "

(by proposition 3.1), E is not A ( 2 f c + e ) . a Remark 4.1.-Notice that the growth "locally" of X^+e (E) for the sets E constructed in propositions 4.3 and 4.4 are "power type" and compare this with proposition 2.2.1.

We now look at some infinite random A(p) sets in Z by considering a method of Erdos. We first introduce a biased coin tossing space on the set of integer sequences Sl (increasing subsequences of N ) . Let X^ be 2-valued random variables (independent) for n > 1, with P(X^ = 0) = 1 — p^ and P(X^ = ! ) = ? „ for 0 <pn < 1 and (pn)^=i a S^611 sequence.

It is natural to call Sl a biased coin tossing space: The probability space on which the X^'s are defined can naturally be taken to be the Cantor set D = { 0 , l}^ . On each factor introduce the probability P»({0}) = 1 -pn and ? „ ( { ! } ) = ? „ . Then the P above is just P = ® P^ and the X^ ' s are the projection onto the nth

n= 1

coordinate. Using the natural identification between S2 and D we have a coin tossing measure on S2.

For different choices of (?„) we get different probability spaces (though by a theorem of Kakutani [16] if ?„ is sufficiently close to p^ for all n, the spaces are the same). We denote a generic sequence of S2 by (a^^ . We always choose (?„) so that 2d p^ = + oo. This insures that the sequence (fl^)^= ^ is infinite

n>\

with probability 1 (by Borel-Cantelli). Recall the following simple variant of the strong law of large numbers (see [13]).

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PROPOSITION 4.5. —Let 0^n)n>\ ^e a sequence of independent random variables on a probability space (S2,F,^). Let S^ = S X,

i<n

and suppose that: 1) E(X,) > 0 2) lim E(S,) = + oo 3)

l-^oo

V Var(X.)

—0^2" <^ + °°- ^^n WY^ probability 1, w^ Afl^

i > l E(S,)

(S^ - E(S^))/E(S^)—^ 0.

As an immediate consequence we have in our case :

PROPOSITION 4.6. -Let (X^^ and Sl be as in our case. If in addition to the previous conditions on (p^)^= i ^e have

S _P"(Lr-P"L_ < + oo

n>l (Pi + . . . +?„)'

?/ie« 0.0. coG S2

X, + . . . + X„ _

1 . Pi + • • • + Pn

It follows that a. a. Ci?€X2 we have

S P,

lim "^-=1

fc-^+oo k v^ith (Of) being a generic sequence.

The essence of the method is that by choosing (p^)^ i carefully we impose a growth rate on almost all sequences by proposition 4.6.

This in turn forces some nice properties to hold.

PROPOSITION 4.7. - Let 2 < / E N , 0 < e < l / / and set 1 — / e c

c=—^——.Let Pfc^o-i/o^ for f c € N . Let ( ^ ) ^ ^ e n be a random sequence. Then wth probability 1 we have

|R((^);Li .l.n) \ < [ l / / e ] except for finitely many n.

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The case / = 2 is classical and due to Erdos and Renyi [8].

The proposition above is proved with some minor modifications to their proof. For a detailed proof in the case of / = 2 see [8]. Let us also note the following consequence of proposition 4.6.

PROPOSITION 4.8. — With the same hypotheses as in proposition 4.7 we have that for each 6 > 0, there exists an 0 < e < I / / s.t. with probability 1 we have a^ ^ k1^ .

Proof. — By proposition 4.6 we have a.a. (a^)^ Sl that

SP<

lim"^———!

fc-*°o k

since we have that

1 ^(i/0-e

pl^'^^pn=CL^^^^CW~e^=nlfl~^e

^ since I y-0"^ ,,< 1: . . pn . < ^ o o since

n>l (Pi + • • •+^ ) n>l (Pi + . . . +Pn)

l-l€

S , . , , , . < 4- oo. So lim —k- = 1 which implies n>i ^l + l / / + c ^+00 fc

a^ ^ k1 by choosing e ==———— . n6

/ ( / + 6)

We can now easily construct A(p) sets (in fact most integer sequences will do) that are not A(p + e). We have the following :

PROPOSITION 4.9. - Let I € N , / > 2 . 5^ p = 21 and let ri > 0. 77?^ a.o. subsequences A = (^)^i , fl^ A(2/) but not A ( 2 / + r ? ) .

Proo/ — By proposition 4.7 and proposition 3.2 a.a. subsequences A are A(2/) and ^ - k1^ for any fixed 6 , 0 < 6 < 7?/2. One may now conclude that A is not A(2/ +17) by using Rudin's proposition on arithmetic progressions (see [23]), but an alternative argument is: By a theorem of Marcinkiewicz and Zygmund (see [27]), [^,. . . , e^}^ ^ /^ for any 1 < q < oo where the constant

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of isomorphism k^ doesn't depend on n. If (a^) is A ( 2 / + 77) then the 2 and 2/ + 77 norms agree on (a^)^^ , but we can pack [^l / / + a] a ^ ' s in 1 , . . . , , n, so arguing as in proposition 2.2.1 we have that,

\i^ ((^)) > ^r?) ^-^w^-

26

)^)

where c(l,rf) is a constant depending on / and 77. Since 6 < r } / 2 we have a contradiction, a Note that the case / = 2 in proposition 4.8 gives a^ ^ k2^6

for any 5 > 0 being A (4). This is a priori somewhat surprising in view of the fact that a^ = n2 is not A(4) (see [23]). One may give a refinement of this result by calculating ^((k2)^^) by using essentially the same techniques as in [23]. We first recall the following result of Landau (see [14]) (and also due independently to Ramanujan (see [14])). This is the only additional result needed in the technique of[23].

PROPOSITION 4.10. -Let ^ G R ^ . Let BQc) be the cardinality of n <; x , n E N which are representable as sums of two squares.

Kx (\ 1 V72

Then B(x)-^^ where K = ( ^ n -^-^) =.764...

p = 3 (mod 4) p prime

PROPOSITION 4.11. — Let n E N . We have for all e> 0 there exists N , E N s.t. \((k2)^^) > (1 - e) c(log n)^ where n>^,

2178

and c = 7T/2 —-,— where K is as in proposition 4.10.

K.

Remarks. — 4.2) It should be noted that while the previous constructions yielded A(p) sets which were not \(p + e) for all e > 0, the construction in proposition 4.9 is not uniform i.e. for a fixed 77 > 0 a.a subsequences A = (a^'^ are A(2/) but not A ( 2 / + 7 7 ) . This may be the price one has to pay for random constructions.

4.3) It should be noted that \((k2)^^) = O(^) for any 6 > 0. This follows from the fact that

IR^2};0^^^)!^^6) for all 8 > 0 (see [15]) and because of proposition 3.2.

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5. Construction in Z(p) C Z(p) C . .. (p > 2).

In this section we will show that for p > m there is a A(2w) set in Z(p) OZ(p) 0 . . . which is not A(2w + e) for all e > 0.

For m > p we don't have a construction but we indicate a possible solution. Prior to this however we need an appropriate analog of proposition 2.2.2. Proposition 3.2 is not useful when most elements have small order, such as in Z(p) ^ Z(p) ( & . . . . One may prove the appropriate generalization of proposition 2.2.2 by suitable modifications of its proof, however we choose to improve the estimates in [6]. One may perform a modification of the proof in [6]. The modification of the proof of proposition 2.2.2 is much more cumbersome. We require the following definition:

DEFINITION 5.1. —Let r be an abelian group and let 2 < ^ e N , 2 < p C N . For A C F denote by Rp (A, n) all functions f: A —> N s.t. /(X) < p - 1 for all X G A and H /(X) = n.

X G A

For 7 e r , R ( A , ^ , 7 ) denotes all f s.t. H X-^ = 7 and

X G A

/GRp(A,M).

PROPOSITION 5.1.— Let G be a compact abelian group with dual group F and assume that A C r has elements only of order p , p > 2 . Also assume | Rp(A,w ,7) I < M for all 7 G F and for all m s.t. 2 < m e N , m < n C N. Then

X ^ A X M ^ f p ] -n)17^!)^.

\ [P\ I

We now turn to the construction of A (q) sets in Z(p)C Z(p)C . . . .

We will try to follow the ideas embodied in section 2. Fix 2 < m G N . Suppose we could show the following for all w G N , n>n(m)'.

there exists A ^ C { 0 , l , . . . , p — 1}^ (which is regarded as an abelian group with the group operation on tuples being coordinate- wise addition mod p ) s.t.

1) \\\>Pn

2) If

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CONSTRUCTION OF A SETS (57

< X n = s u p { \ R p ( \ , k , x ) \ \ x ( = { 0 , . . . , p - ! } ' " " , 2 ^k<m}

then sup a^ < + oo . n

Then identifying the dual group of Z^)" with finite tuples of 0 ' s , . . . , p — 1 ' s as in section 2 ; we way construct the required types of A(p) sets. This is done by patching the sets A,, as in section 2. |A,, | > p" and proposition 3.1 ensures that the patched set is not A(2m + e) for all e > 0 and sup a,, < + oo ensures using proposition 5.1 that the A^'s have a umform (in n)A(2m) constant (and thus a union of A/s is A(2m); with the A^'s built on "disjoint blocks" of Z(p) e Z(p) ® . . . ) .

Proposition 2.1.1 suggests that A^ may be built by considering a suitable choice of (^, £ N" and letting,

A,, = { ( xk l, . . . , xk m) ^ x e G F ( p " ) }

with 1 = k, < k^ < .. . < k^ (we regard GF(p") as a vector space over GF(p) and expand x ' in a basis expansion and put in the coordinates in place of x " ' , so that A,, C { 0 , . . . , p — I }"•").

The choice of ^i = 1 insures that | A^ | = p" . (Actually any choice for k^ will work so long as x —> x "¹ is an automorphism for GF(p") (fixing GF(p)). Since the automorphisms form a cyclic group of order n, the most "convenient" choice is k^ = 1). The main problem is making sure that sup a < + oo.

n

In view of the above remarks it is easy to see that we may reduce to the following problem on Diophantine equations over finite fields:

Problem 5.7. - Let 2 < m € N and p be a prime which are fixed. Let n e N, {^ , . . . , y^} c GF(p") and

1 = k, < ^ < . . . < ^

with the k, e N for all i. Consider the system of equations :

*i . , *i

^m

C")

^, + • • • + ^ ^ y ,

^ + . . . + ^ =y,

x,km+...+ x^ =^

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By a "solution" to this system we mean an ( x ^ , . . . , x^) E GFQ^)^"

which satisfies (*) and for which the number of non-zero x / s which are the same in ( X i , . . . , x^) is at most p — 1. Denote by g(n , y ^ , . . . , y^ , k^ , . . . , k^) the number of solutions to the above system (*). The question then is: Can one find one fixed set of k i ' s , 1 = k^ <k^ < . .. <k^ s.t.

g(n,y^ . . . ,y^ ,^, . . . , k^) <C

for some C G N uniformly in n and {y^ . . . , ^}? Of course one may replace uniformity in n by working in the algebraic closure A = U GF(p") and considering the equivalent problem of

n> 1

getting a bound on the number of solutions (to (*) of the type (x i , . . . , x^) G A'" (with no more than p — 1 of the non-zero x / s being the same)) uniformly for [y^ , . . . , y^} C A .

We now show that at least for p > m, one may solve the above problem quite simply.

PROPOSITION 5.2. — // p>m and one chooses k ^ = i , I < i^m then g ( n , y ^ , . . . , y^ , 1 , 2 , . . . , m) < m !

Proof. — Fix n € N . The condition that at most p — 1 of the non-zero x/5 in (^ , . . . , x^) ( ( x ^ , . . . , x^) being a solution of (*)) are the same is automatic since p > m. We will show that the assumption,

m 2m

L ^ = ^ x\ for / = 1 , . . . , m

( = 1 i=m+l

implies that set of x ^ ' s , 1 < / < m counting multiplicity is the same as the set of x^s m -t- 1 < i < 2m counting multiplicity. Let Y\, ' - - , Ym ^e indeterminates adjoined to GF(//1). Denote by

k

S k ( V i . • • • . Yk) = ^ Y^ and by a ^ , . . . , a^ the elementary 1=1

symmetric functions in y ^ , . . . , y^ . By Newtons's identities (see [25]) the a / s can be written as polynomials in the S j ^ ' s since p>m.

It follows that a^x,, . . . , x^) = a,0c^ , . . . , x^) for i = 1, . . . , m. So

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( X - ^ ) . . . ( X - x ^ ) = = ( X - ^ + i ) . . . ( X - ^ )

for all X € GF (p"). So some ^. (1 < i < w) is the same as an Xj Xj(m + 1 < / < 2 w). By relabelling the x ^ s we can have x^ = x^ . Cancelling these from the equations

m 2m

^ x7 = ^ ^ , / = 1 , . . . , m ,

, = = 1 ' i = w - H

we may repeat the above argument with the equations

m - 1 2 w -1

Z ^ = S ^ y = = l , . . . , ^ - l / = i i = ^ i

(the equation with f = m is ignored since it is of no further value) to conclude that x^_^ = ^ 2 w - i a^ter relabelling the x / s . The procedure can be continued to terminate the argument, n Proposition 5.2 therefore gives the required construction for A ( p ) sets for p>m. For m^p the problem 5.1 is much more difficult and no solution seems to be known. The following remarks are due to P. Deligne.

Remarks. — 5.1) For a fixed set of y ^ s , [ y ^ , . . . ,j^ } C A (see problem 5.1) if the variety of solutions consists of isolated points

m

then by Bezoufs theorem they are at most n kj of them. For i

Bezoufs theorem see [20].

5.2) If there are infinitely many solutions to (*) on the algebraic closure A then there is a formal power series :

00

(X,(0)=X(0 : X ( 0 = S ^^w ^

k=0

which is formally a solution with x ^ ^ O (i.e. one of x ^ ^ ^ O ) and

1)^/0^/ =yf

i

2) For all k>0, the coefficient of ^ in ^ X,(r/7 is 0.

i

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No use has been made of the above however.

5.3) One may weaken the problem (since the reader will observe this is all that is really required for the construction) by requiring only that g(rif, ^i , . . . , y^ , k^ , . . . , k^) < C uniformly for some sequence of sets of k ^ s (i.e. the set of m k / s is allowed to change with n^ however k^ = 1 for all /'), some subsequence (^.) of N s.t. ^ ——> +00 and for all { ^ , . . . , y ^ } C GF (p^').

However Professor Deligne thinks that solving the weaker problem for a thin set of n^s does not really help much and presumably a solution of the weaker problem will in fact enable one to solve problem 5.1.

6. Construction in Z (p°°).

We construct a A (4) set in Z(p°°) which is not A ( 4 + e ) for all e > 0 . Some results are also possible for A ( p ) sets, p>4. The construction will be done by showing the existence of sets

E , ^ Z ( p ^ ) + x ,

(for some nj, ——> + oo , some Xj, € Z (p°°)) , ^ = 1 , 2 , . . . such that 1) | E^ | >c p"^2 (c independent of k)

2 ) | R ( U E ^ , 2 , 7 ) 1 < 1 for all 7 ^ Z ( p ° ° ) .

k

By the remark 3.1 and (1) it follows that U E^ is not A (4 + e)

k

for all e > 0 and pro position 3.2 and (2) insure that UE^ is A(4). The

k

idea of the construction is that by proposition 4.2 it is easy to construct F^ satisfying (1) and having 2-sums out of F^ being distinct.

The sets E^ (are modified F^s) are constructed by induction. The main tool will be a well-known theorem of Turan's in extremal graph theory [3].

PROPOSITION 6.1. — The maximal graph on n vertices without an l-clique is achieved by splitting the n vertices into /-I sets of

n n 1

cardinality -——- and -——-\ 4-1 and placing an edge between

any pair of vertices in different sets. If

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n==(l - l)m+r, 0 < r < / - 1 then the cardinality of the number of edges of this graph is

•<)-.('"; ')-.-i-<).

The following proposition follows trivially from proposition 6.1.

PROPOSITION 6.2. - Let S be a set with | S | = n. If A C P, (S) (a// 2-subsets of S) w^A IM^"^" n _ (-" _ i ) ^^

A D P ^ B ) /or B C S wfA | B | > " .

Proof. - We think of S as the vertices of a graph G where an edge {a, b} is in G if and only if [a, b} e A. We show that if A has cardinality at least as much as above then G has a n/2 clique.

Case 1 : If n = 2k for some k, then set l = k + l , m = l , r=0, so that n = (I - 1) m + r. If

I A I > 1 + ( : ; ) - ^ ) = 1 ^ ( . - 2 )

then there exists a B s.t. | B | = w / 2 + 1 and A3P,(B) by proposition 6.1.

Case 2 : If « = 2 A ; + 1 for some k write n = ( l - l ) m + r with l = k + l , m=2, r= 1.

By proposition 6.1. if | A | > — — — — - 1 then there exists B with | B | = "—— and A D P^ (B).

So the result follows by comparing case 1 and case 2. D Remark 6.1. - If S = { 1 , 2 , . . ., 2 n} then put

A = {{a, b} \Ka<n, n+ Kb<2n} . n2

Then I A | = ^ - yet A ^ P, (B) for any B with 1 B | > 3 . This

shows the sharpness of proposition 6.1.

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Remark 6.2. - If \S\=n and n>fi(q) and A C ? 2 ( S ) is such that | A | > ^ ^ "(9) then ^ Kamsey's theorem A D P^ (B) for some B with | B | = q . Unfortunately q is only about log n (see [12]). For our application we need the size of B to be propor- tional to n.

We now start the inductive construction. Since some of the details are cumbersome, we shall not give all details especially if it is obvious as to what to do.

PROPOSITION 6.3. - There is a A (4) set in Z(p°°) which is not A ( 4 + e ) for all e>0.

Proof. - Assume E^ , . . ., E^ have been constructed with the two properties discussed above (and so | R ( U E., 2, 7) | < 1 for

v / < fc /

all 7 G Z O?00)). Denote E = U E, and let E' = E^ ^ which

f<k

is to have the properties (and will be constructed below) that EtC Z ( pn k + l) + x ^ ^ for some ^ + i ^ = N ( ^ ^ > ^ ) and x ^ + i eZ(p°° ), \Ef\>cpnk+l/2 and | R(E U E', 2 , 7) | < 1 for all 7 E Z ( p0 0) . First choose F^ C ZQ^ ), ^.^ = m > n^ (m is much bigger than n^ , we will choose m so big that a number of properties will be satisfied. We leave its size unspecified because it will be-clear that such an m will exist), and so that \ F ^ \ > c pm / 2

(all constants in this proof are denoted by c and are independent of n^) and finally that 2-sums from F^ mod^^ are distinct according to proposition 4.2.

Considering the possible interactions between E and F^ we have 3-cases which may violate R (E U F ^ , 2, 7) < 1 for all 7 G Z (p°°)

Case 1 : a + b' = c + d ; [a , b\ c } C F , , d E E { a , b ' } ¥= { c ' , d } .

C a ^ 2 : ( a ) a ' + & ' = c + ^ ; [a , b ' } C F ^ , { c , d } C E [a , b ' } ^ { c , d } , and (b) a 4- b = c + d ;-{a\ c } C F ^ , { 6 , d } C E .

Case3 '. a ^ b = c + d and a E F^ {b, c, d} C E, {a\ &}^ { c , r f } .

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We discuss each of these 3 cases separately. Our starting out assumption will be that E H F ^ = 0 . We may assume this because m may be chosen so large that | F^\E | > c p17112. F^ = F \E will be the new F ^ . All primed elements from this point will be from our possible new set and unprimed ones from the old set.

Case 1 : We want to avoid a' + b' = c + d , [ a ' , b1}^ {c, d\ It is enough to have (F^ + F^ - F^) n E == 0. By choosing n large enough E C Z ( pn) . Assume m much larger than n and F^ C Z ^ p ' " ) with | F^ | > cp^2 . Choose n' larger than m and pick

x € Z ( pn f) \ Z ( pm) . Set F3 =F, +;c.

Then (F3 + F3 - F^) n E = 0, otherwise ;c E Z (^ ). Also

^ 3 1 = ^ 1 , F3 C Z ( ^ ) + ; c

and 2-sums out of F^ are still distinct in Z(p°°). Fg will be the new F^ . Of course we may assume F3 n E = 0 by taking m large and considering Fg\E if needed.

Case 3 : We want to avoid a + b = c 4- d with {a', &}=^ { c , rf}.

This time we want (F3 n E + E - E) = 0. -It should be clear to the reader that the translation technique of case 1 will again work, giving a new set F ^ C Z ( pm) ^ y for some y E Z (p°°). We should remark that when using the translation technique in succession we might have to translate using elements of larger and larger Z ( pn)ls so as not to cancel the effect of previous translations. Again we may assume that

F^ n E = (Z).

Case 2 ( a ) : We want to avoid a ' + b ' ^ c + d . This time we want (F^ + F4) n (E + E) = 0 . Again this can be handled as above with a new set F^ C Z (p771) + z for some z G Z (p°° ).

Case 2 ( b ) : Now we want to avoid a 4- b = c 4- d with [ a , b}^ {c . d}. The fact that Fg n E == 0 (which we may assume) and {a , b} ^ {c , d} means that we may assume a , b, c , d are all different. It is enough to show that

((FS - Fs)\{0}) n (E - E)\ {0}) = 0 . Since differences in F^ are unique, if we set

G = ( F 5 - F 5 \ { 0 } ) \ ( E - E ] { 0 } ) then by choosing m large we can have (G | as "close" to

| F s - F 5 \ { 0 } |

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as we want. Define a map 0 :.(F5 x Fg)\A —> (F^ — F5)\{0} where A = { ( / , / ) I/O FS } by 0 (a', b ' ) = a - b ' . Then 0 is bijective.

Define ^ : Fs x F s \ A — — ^ ( F s ) by ^ (a , b ' ) = {a', b ' } . Then

^ is surjective with |^"-1 ( { a \ &7}) | = 2 for all {a', &'} € P^ (Fg).

Let A = ^ 0 -1( G ) . Then | A | is as "close" to IP^Fg)! as we want (by choosing m large). If we know there exists

B C FS s.t. | B | > c | Fs I and P^ (B) C A then we would be done by setting F^ = B. That this can be done is guaranteed by proposition 6.2 (by choosing m large enough).

Now ^k+i = ^6 anc^ m=:nk+l worlcs. n For A ( p ) sets with p > 4 we have some troubles. This is because neither the translation technique nor the technique of case 2(b) (suitably generalized) helps in dealing with the case e.g. when we want to avoid a' + b' + c = e -I- /' + g (using the notation of the proof) in the A (6) case. Translation obviously doesn't work and to use the other technique would mean that 4-sums from our new set (which we want to adjoin) would have to be distinct (or at least meet the requirement of proposition 3.2) in which case we only would have the new construction being not A (8 + e). In any case it is at least possible to show :

PROPOSITION 6.4. - // 2 < / : G N then there is a A (2k) set v^hichisnot A ( 4 A : - 4 + e ) for all e> 0 in Z(p°°).

Let us also note that from the work of the previous sections we have as a particular consequence :

THEOREM 6.5.—For any compact abelian group G there is a A (4) set in G* v^hich is not A (4 4- e) for all e > 0.

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CONSTRUCTION OF Ap SETS 165

BIBLIOGRAPHY

[I] G. BACHEUS and S. EBENSTEIN, On A(p) sets, Pacific. J. Math., 54 (1974), 35-38.

[2] G. BENKE, An Example in the Theory of A(p) Sets, Bollettino U.M.I., (5) 14-A(1977), 506-507.

[3] B. BOLLOBAS, Graph Theory, An Introductory Course, Graduate Texts in Math, Vol. 63 (1979).

[4] A. BONAMI, Etude des Coefficients de Fourier des fonctions de lf(G), Ann. Inst. Fourier, Grenoble, 20, fasc. 2 (1970), 335-402.

[5] R. BOSE and S. CHOWLA, Theorems In The Additive Theory of Numbers, Comment. Math. Helv., 37 (1962/1963), 141-147.

[6] R. EDWARDS, E. HEWITT and K. Ross, Lacunarity for Compact Groups I, Indiana University Math. Journal, 21 (1972), 787-806.

[7] P. ERDOS, Problems and Results in Additive Number Theory, Colloque sur la Theorie des Nombres, Bruxelles (1955),

127-137.

[8] P. ERDOS and A. RENYI, Additive Properties of Random Sequences of Positive Integers, Acta. Arith., 6 (1960), 83-110.

[9] P. ERDOS and J. SPENCER, Probabilistic Methods in Combinatorics, Academic Press, 1974.

[10] T. FIGIEL, J. LINDENSTRAUSS and V. MILMAN, The dimension of Almost Spherical Sections of Convex Bodies, Acta. Math., 139 (1977), 53-94.

[ I I ] C. GRAHAM and O.C. MCGEHEE, Essay sin Commutative Harmonic Analysis, Springer-Verlag, 1979.

[12] R. GRAHAM, B. ROTHSCHILD and J. SPENCER, Ramsey Theory, Wiley Interscience, 1980.

[13] H. HALBERSTAM and K. ROTH, Sequences, Oxford University Press, 1966.

[14] G. HARDY, Ramanujan, Chelsea, 1959.

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[15] G. HARDY and E. WRIGHT, An Introduction to the Theory of Numbers, Oxford, 1938.

[16] S. KAKUTANI, On the Equivalence of Infinite Product Measures, Ann. of Math., 49 (1948), 214-224.

[17] J. KOLMOS, M. SULYOK and E. SZEMEREDI, Linear Problems in Combinatorial Number Theory, Acta. Math. Sci. Hungar., 26 (1975), 113-121.

[18] J. LINDENSTRAUSS and L. TZAFRIRI, Classical Banach Spaces I , Springer-Verlag, 1977.

[19] J. LOPEZ and K. Ross, Sidon Sets, Marcel Dekker, 1975.

[20] D. MUMFORD, Introduction to Algebraic Geometry, Harvard Lecture Notes, 1967.

[21] J. ROTMAN, The Theory of Groups, Allyn - Bacon, 1973.

[22] W. RUDIN, Fourier Analysis on Groups, Interscience Publishers, 1962.

[23] W. RUDIN, Trigonometric Series With Gaps, J. Math. Mech., 9 (1960), 203-227.

[24] H. RYSER, Combinatorial Mathematics, Cams Mathematical Monographs 14, Mathematical Association of America, 1963.

[25] B.L. VAN DER WAERDEN, Modern Algebra, Ungar, 1953.

[26] J.H. VAN LINT, Introduction to Coding Theory, Graduate Texts in Mathematics, 86, Springer-Verlag, 1980.

[27] A. ZYGMUND, Trigonometric Series, Cambridge University Press, 1959.

Manuscrit re^u Ie 30 septembre 1983 revise Ie 23 aout 1985.

D.J. HAJELA,

Bell Communications Research 435 South Street, MRE 2P-372 Morristown N.J. 07960 (USA).

Construction techniques for some thin sets in duals of compact abelian groups

D. J. H AJELA

Construction techniques for some thin sets in duals of compact abelian groups

CONSTRUCTION TECHNIQUES FOR SOME THIN SETS IN DUALS OF COMPACT

ABELIAN GROUPS

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