Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3:August 1995, pp. 281 285. �9 Printed in India.

A note on the growth of topological Sidon sets

K GOWRI NAVADA School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

MS received 31 October 1994; revised 14 December 1994

Abstract. We give an estimate for the number of elements in the intersection of topological Sidon sets in R" with compact convex subsets and deduce a necessary and sufficient conditions for an orbit of a linear transformation of R" to be a topological Sidon set.

Keywords. Topological Sidon sets; growth of sets.

Given a locally compact abelian group G, a subset A of the dual group X is called a topological Sidon set if any b � 9 ~ (A, namely any bounded complex-valued functions on A, is the restriction to A of the Fourier transform of a complex bounded Radon measure on G. These sets play an important role in harmonic analysis ([LR], [M]). When G is compact, X is discrete and the notion of topological Sidon sets coincides with that of Sidon sets. ([LR], [M].)

For any topological Sidon set A as above there exist c/> 1 and a compact subset K of G such that any be l ~ (A) is the Fourier transform of a measure which is supported on K and has norm at most c Ilbll ~. When this condition holds for a c/> 1 and a compact subset K, A is called a (c, K) topological Sidon set.

Sidon sets are known to be 'thin' set ([LR], [M], [P] ). Further, estimates are known for the number of elements in intersections of Sidon sets with finite subsets (see Theorem 3). The purpose of this note is to give the similar estimate for the number of elements in intersections of topological Sidon sets in R = with compact convex subsets. Let I denote the Lebesgue measure on R ' . For a set E we denote by I EI the cardinality of E. Then our result shows in particular the following.

Theorem 1. Let m e N . Then for any compact set K c R m and c >>, 1, there exist a d > 0 and a neighbourhood U o f 0 �9 R m such that for any (c, K) topological Sidon set A of R m and any convex subset A o f R m we have

lanai ~< dlog( l (A + U)fl(U)).

We deduce from the theorem the following criterion for orbits of linear transform- ations to be topological Sidon sets.

COROLLARY

Let A:Rm~-*R = be a linear transformation and v e R m. Then {A"(v) lneN} is an infinite topological Sidon set i f and only if v is not contained in any A-invariant subspace of R m on which all the eigenvalues are o f absolute value at most 1.

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282 K Gowri Navada

While the estimate as in the theorem is adequate for the above corollary, it seems worthwhile to note that our argument below gives not just existence of a neighbour- hood U, but a concrete way of choosing such a neighbourhood. This is of some interest since the right hand side would typically be big when U is small and so for getting a better estimate one would be interested in choosing U as big as may be allowable. We shall prove the following stronger version of theorem 1.

Theorem 2. Let m e N . Then for any c >1 1 there exists a d > 0 such that the following holds: for any compact set K of R m, any(c, K) topological Sidon set A of R m and any convex subset A of R m we have IAc~A[ ~< dlog(l(A + 3U)/I(U)), where U = {2eRmlsupx~K~n 1~= 12ixi[ <~ 1/4rtc}, B being any basis o f R".

We shall now recall a result from [LR], on which our proof of Theorem 2 is based, prove some preparatory results and then proceed to prove the theorem.

A finite subset A of a discrete topological group X is said to be a test set of order M; where M/> 1, if [A2 A - x I <~ MIA[.

Theorem 3 [LR]. I f E c X is a Sidon set with Sidon constant x >1 1, then l A n E [ <~ 2x2eM logl A[ for test sets of order M such that ]AI ~> 2.

The following proposition signifies that any countable set close to a topological Sidon set is again a topological Sidon set. It is just a higher dimensional version of Lemma 3 of Ch. VI of [M] and is deduced analogously, as indicated below.

P R OP OS I TION 1

Let A = {2.}~: 1 be a (c, K) topological Sidon set in R m and p > 1 be given. Let e be such that (1 - ec)- 1 = p and let 0 < 0 < e and W = {2eR~lsup~xli~ 12ixl[ <~ 0/4n}. For each n let 2'.e2. + W. Then A ' = {2',}.% 1 is a topological Sidon set and further any function b in I~(A ') is the restriction to A' of the Fourier transform of a measure p eM(R m) with

!1/~11 < pcllbll o0.

Proof. SinCe A is a topological Sidon set, A is a coherent set of frequencies. (cf: [M], Theorem I ofCh. VI for a proof in the case m = 1. The proof actually holds in general.) We now argue as in the proof of the assertion (a)=*-(c) in Theorem X of Ch. IV of [M]: The argument there shows that for the set W as above {2. + W}.~ 1 are mutually disjoint and if H : A + W - ~ A x W is the (well-defined) map such that H(2. + u) = (2., u), for all neN, ue W, then for each g~B(A x W), g oH~B(A + W) and

llg~ < P [Iglln(A x w)" ~ Let b~l~176 ') be given. Let f (2 . + u) = b(2'.), Vn, Vu~ Wand let f be the restriction of

f t o A. Since A is a (c, K) topological Sidon set, there exists a measure p~M(R m) such that a=f on A and Ilpll ~< cllflloo. Then/~ x 6 o yields an element of B(A x W); we denote it by g and put ~ = goHeB(A + I40. Then

Ilvlln(^§ w~ < p IIglIB(A • w) ~< P 11/~ X 60 II ~< pc Ilfllo = pc [Ifll ~.

Hence there exists a measure v e M ( R ' ) such that II vii < pc II f II | and g = f o n A + W;in particular IIv[I < pcllbllo~ and ~ = b on A'.

Let {x 1 . . . . . x,.} be any linearly independent set in R" with m elements. Then any translate of the set {Ei%lqxi[0 ~< ti ~< 1} is called a parallelopiped in R ' ; further if {xl . . . . . xm} is an orthogonal set, then such a parallelopiped is called a box.

Growth o f topological Sidon sets 283

P R O P O S I T I O N 2

Let A be a compact, convex subset o f R ~ with nonempty interior. Then A contains a parallelopiped P such that l(A) ~ (2m)ml(P).

Proof. By a suitable translat ion, we can assume tha t 0cA. We define an o r thogona l set {x~ . . . . . x~ } in R m and a linearly independent subset {Yl . . . . . y , } of A by induct ion as follows. Let x 1 = y l e A bc an element of m a x i m u m norm. Assume tha t for some k ~< m - 1, an o r thogona l set {x I . . . . . x k } and a linearly independent subset {Yl . . . . . Yk } are chosen. Let Pk'Rm- '*(Xl . . . . . Xk~ • be the o r thogona l project ion m a p on to the subspace of R ~ o r thogona l to {x 1 . . . . . x k }. Choose x k + 1 to be an clement of m a x i m u m norm in Pk(A), since A has noncmpty interior Xk+ 1 :/:0. Let Yk+leA be such that Pk(Yk + 1 ) = Xk + 1" Clearly {x I . . . . . Xk + 1 } is an o r thogona l set and {Yl . . . . . Yk + 1 } arc linearly independent . By induction this yields the sets {x 1 . . . . . x m } and {Yl . . . . . Ym } as desired.

Let I be the box generated by {x I . . . . . x,~}, i.e. l={Z,i%it~x~lO<<.ti<~l}. Let J = {Y . i%l t ix i [ - 1 <. t i <. 1}. Then l(J) = 2ml(l). I f a e A and (a I . . . . . am) are the coordi- nates of a with respect to the vectors x I . . . . . x s , then lail ~ IIxil[Vi and hence aeJ . Therefore A __q J and consequent ly l(A)<~ l ( J )= 2m/(l). Let P be the paral lc lopiped generated by {y l /m . . . . . y~/m}, i.e. P = {Z~': 1 t~yi/m[O ~< t~ ~< 1 }. Since A is convex and 0 c A it follows that P _~ A. The matr ix of the t rans format ion x~,--~y~ is lower t r iangular with diagonal entries equal to 1. Therefore l(P) = m m I(I). Hence we get l(A) <~ (2m)"l(P).

Proof of Theorem 2. Wri te A = {2n},~ 1 . Let B be any basis of R m. Let 0e(0, 1/c) be arb i t ra ry and let ee(0, 1/c). Let U 0 = {2eRmlsupx~runlZi% 12ixi[ <~ 0/4n}. We apply Propos i t ion 1 to p = (1 - e c ) - 1 and e, 0 and U 0 as above. Clearly, U s is a convex, compac t and Symmetric nc ighbourhood of 0. Applying Propos i t ion 2 to U 0, we get a paral lelopiped P _ U 0 such that l(Ue)<~(2m)ml(P). Let {z I . . . . . z~} be such that P is a t ranslate of {Ztizi[0 ~< t i ~ 1}. Let L be the lattice genera ted by {z I . . . . . zm}.

! t P oO If wc choose ),,~(2, + U s ) n L, then A - {2,},= l is a coherent set of frequencies with respect to (1, F), where F is a fundamenta l doma in of the annihi la tor L ~ of L. By Propos i t ion 1, A' is a topological Sidon set and any b e L ~ ( A ') is the restriction to A' of the Four ier t rans form of a measure p e M ( R m) with II ~ [I ~ pc IIb II ~o. This implies that A' is a (pc, F) topological Sidon set ( [M]) . Since L = Rm/L ~ and F is a fundamenta l domain for L ~ in R ' , this is equivalent to saying tha t A' is a Sidon set in L with Sidon constant pc.

N o w let A be a compac t , convex subset of R ' . Pu t A + Uo = B and B + Us = C. We shall prove tha t C n L is a test set with associated cons tant (18m)' . We have

I ( C n L ) + (Cc~L) - (Cc~L)I ~</(C + C - C + Uo)/I(P )

l((C + Uo) + (C + Uo) - (C + Uo))fl(P).

C + U o is a convex, c o m p a c t subset of R " with n o n e m p t y interior. Applying Propos i - t ion 2 we get a para l le lopiped PI c C + Uo such that

l((C + Uo) + (C + Uo) - (C + Uo)) <. (6m)' l (Px). Then

(6m)Sl(P~) <. (6m)ml(C + Us) = (6m) ' l (B + Us + Us) <. (6m)m3'~l(B),

because B contains a t ranslate of U s. These inequalities and the fact tha t U s contains

284 K Gowri Navada

P yields that

I(C c~ L) + (Cc~L) - (CnL)I ~< (18m)ml(B)/l(P)

<<, (18m)mlLc~(B + Uo)l = (18m)mlCc~LI �9

This proves that C n L is a test set as claimed. By applying Theorem 3 to C c~ L we now get that

IAc~AI ~< IAn(A + Uo)[ ~< [A'n(A + 2U0)l ~< dl log lLn(A + 2U0)l,

where d 1 = 2e(pc)2(18m) m. Then

IA c~AI ~< dl log(l(A + 2Uo + uo)/l(P)) <~ dlog(l(A § 3Uo)/I(Uo))

where d is a constant depending on c, e and m. By letting 0--* 1/c we get the required result.

The following theorem is analogous to Theorem II in Ch. VI of [M].

Theorem 4. I f {2. }.~ 1 is a sequence in R m such that for some �9 > 1 we have for all large n, 112.+ ~ II t> �9 II 2. I[ then {2.}~= 1 is a topological Sidon set.

This can be deduced from the following lemma in the same way as Theorem II in Ch. VI of [M] from the analogous lemma there.

Lemma. I f {2.}.~ 1 is a sequence in R m such that [12n+ 1 II ~ 6112.11, Vn, and if {b.}~= ~ is any sequence in T, then there exists a point s e R m such that Ilsll ~ 1/1[2111 and [(s, 2 , ) - b.I -,,< 1,Vn.

Proof. Let 2 = (al . . . . , am) be a nonzero element in R m. Let B be a ball in R m with radius 1/]1211 and centre at x o. Let fl=(ax/[[2ll 2 . . . . . am/ll2l[2). Then the points x o +f l are contained in the boundary of B and each of the two line segments joining Xo to x o + fl is mapped onto T by the map x--, (x, 2). Therefore given any b e T we can find a point y e B such that (y, 2) = b and B(y, 1/2 ][ 2 II) -~ B. By induction we choose balls B. and points y . ~ B . such that B.+ 1 c B(y. , 1/6 [{ 2. II ) c n., V n as follows: Let B 1 be the ball with centre at 0 and radius= 1/1121 [I. Let y l e B ~ be such that ( y ~ , 2 1 ) = b l and B(y1,1/611211l)cnl . Suppose B. and y. have been chosen satisfying the above conditions. Let B.+t be the ball with centre at y. and rad ius= 1/112.+11 ]. Then B. + 1 ~ B(y. , 1/6 I[ 2. I[) - B.. Choose y. + i e n . + 1 such that (y . + 1,2, + 1 ) = b. + 1 and B ( y . + ~ , l / 6 l l 2 . + l l [ ) c B . + ~. Let s be the point of intersection of {B.}. Then se c~ o~ B(y. , 1/6 [[ 2. II) also. Then for all n, II s - y . II ~< 1/61[ 2.11 and hence I (s, 2 . ) - b. [ = [(s, 2 . ) - (y . , 2.)1 ~< 1; which proves the lemma.

Proof of the Corollary. There exists a unique largest A-invariant subspace V of R m such that all eigenvalues of A on V are of absolute value at most 1. Suppose ve V. Using Jordan decomposition it is easy to see that there exists a c > 0 such that II A"(v) [I ~< cnm - 1 for all n. Let r. = cn m- t and B. the ball with centre at 0 and radius r.. If {A"(v)}~: 1 = A is an infinite topological Sidon set then A"(v), n e N , are all distinct and hence by Theorem 2 above, we have, n ~< {B.nAI ~< dlog( l (B. + 3U)/l(U)) for some compact neighbourhood U of 0. Therefore there exists a constant D such that n ~< Dlogr . for all n. Since r. = cn m- 1 this implies that n/logn is bounded which is a contradiction.

Growth of topolooical Sidon sets 285

Now suppose that vr V. Using Jordan decomposition one can see that there exists a c > 1 and an integer k ~> 1 such that II A" + k(v) II >t C II a"(v) II, for all large n. It follows from Theorem 4 that A is a finite union of topological Sidon sets. Since A is uniformly discrete it is a topological Sidon set.

Acknowledgements

The author thanks Prof. S G Dani for suggesting the problem and for many helpful discussions and also wishes to thank the National Board for Higher Mathematics for the financial support. Thanks are also due to the referee for useful comments enabling improvement of the text of the paper.

References

[LR] Lopez J M and Ross K A, Sidon Sets, Lecture Notes in Pure and Applied Mathematics, 13; (New York: Marcel Dekker)

[M] Meyer Y, Algebraic numbers and harmonic analysis (Amsterdam, London: North-Holland Publishing Company) (1972)

[P] Pisier Gilles, Arithmetic characterization of Sid0n sets; Bull. Am. Math. Soc. 8 (1983), 87-89