Propriétés de décomposition pour les ensembles de Sidon. (French) Zbl 0546.43006

Bourgain, J.

Bull. Soc. Math. Fr. 111, 421-428 (1983).

The paper consists of two parts. The main result of the first is the following: if \(G_ 1\) and \(G_ 2\) are compact abelian groups and \(\Lambda\) a Sidon set in \((G_ 1\times G_ 2)^{\wedge}\) then \(\Lambda\) is a finite union of sets \(\Lambda_{\alpha}\) such that, for each \(\alpha\), the projections of \(\Lambda_{\alpha}\) on \(\hat G_ 1\) and on \(\hat G_ 2\) are injective and both are Sidon sets in \(\hat G_ 1\) and \(\hat G_ 2\) respectively. As a corollary the author derives the following extension of the known theorem of Malliavin-Brameret and Malliavin: for any square-free n, every Sidon set \(\Lambda\) in the direct sum \(\sum {\mathcal Z}(n)\) is a finite union of quasi-independent sets \(\Lambda_{\alpha}\), this means that there is no relation of the form \(\sum_{\gamma_ j\in\Lambda_{\alpha}\backslash\{0\}}\epsilon_ j\gamma_ j=0 (\epsilon_ j=\pm 1)\). The method of proof is based on the probabilistic characterization of Sidon sets by Rider and on some related results. In the second part the author calls a set \(A\subset\hat G \ell\)-quasi-independent if any relation \(\sum_{\gamma\in A}\epsilon_{\gamma}\gamma =0\) with \(\epsilon_{\gamma}\in\{-1,0,1\}\) and \(\sum| \epsilon_{\gamma}|\leq \ell\) is trivial and he proves that, for each \(\ell\), every Sidon set is a finite union of \(\ell\)-quasi- independent sets. He begins by proving a combinatorial theorem, interesting in itself, which reads: let \(S_{\ell}(A)=\{\sum_{\gamma\in A}\epsilon_{\gamma}\gamma:\quad\sum |\epsilon_{\gamma}|\leq \ell\};\) there exists a function \(\Phi\) (\(\ell,C)\) such that if \(\Lambda \subset\hat G\) and \(| S_{\ell}(A)\cap\Lambda |\leq C| A|\) for any finite set \(A\subset\Lambda \), then \(\Lambda\) is a finite union of \(\Phi\) (\(\ell,C) (\ell +1)\)-quasi-independent sets. Since this theorem applies in particular to Sidon sets, the above-mentioned result follows. As a corollary one obtains the statement that every Sidon set is a finite union of sets tending to infinity (for the definition see the book of J. M. López and K. A. Ross [Sidon sets (1975; Zbl 0351.43008)], where this statement is proved for countable Sidon sets and left as an open question for the other ones).

Reviewer: S.Hartman

MSC:

43A46

Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

Keywords:

compact abelian groups; Sidon set; quasi-independent sets

Citations:

Zbl 0351.43008

References:

[1]	FERNIQUE (X.) . - Régularité des trajectoires des processus gaussiens , École d’Été de St.-Flour, Springer LNM 480. Zbl 0331.60025 · Zbl 0331.60025
[2]	LINDAHL (L. A.) , POULSEN (F.) . - Thin sets in harmonic analysis , Lect. Notes in Pure and Applied Math., M. Dekker Inc., New York, 1971 . MR 52 #14800 | Zbl 0226.43006 · Zbl 0226.43006
[3]	LOPEZ (J. M.) , ROSS (K. A.) . - Sidon sets , Lect. Notes in Pure and Applied Math., M. Dekker Inc., New York, 1975 . MR 55 #13173 | Zbl 0351.43008 · Zbl 0351.43008
[4]	MALLIAVIN-BRAMERET (M. P.) , MALLIAVIN (P.) . - Caractérisation arithmétique des ensembles de Helson , C.R.A.Sc. Paris, Série A, 264 ( 1967 ), 192-193. Zbl 0154.02401 · Zbl 0154.02401
[5]	MARCUS (M. B.) , PISIER (G.) . - Random Fourier series with applications to harmonic analysis , Annals of Math. Studies n^\circ 101, Princeton University Press ( 1981 ). MR 83b:60031 | Zbl 0474.43004 · Zbl 0474.43004
[6]	PISIER (G.) . - De nouvelles caractérisations des ensembles de Sidon , Advances in Maths, Supplementary studies, Mathematical Analysis and Applications (Part B), vol. 7, 1981 . MR 82m:43011 | Zbl 0468.43008 · Zbl 0468.43008
[7]	PISIER (G.) . - Conditions d’entropie et caractérisations arithmétiques des ensembles de Sidon , preprint. · Zbl 0539.43004
[8]	VAROPOULOS (N. Th.) . - Some combinatorial problems in Harmonic Analysis , Summer School in Harmonic Analysis, University of Warwick, 1968 .