[Sur les ensembles de sommes et produits multiples d'ensembles finis d'entiers]
Soit un ensemble fini d'entiers et |A|=N⩾2. Pour tout entier positif k, denotons kA (resp. A(k)) l'ensemble de toutes les sommes (resp. produits) de k éléments de A. On démontre que pour tout b>1, il existe k=k(b) tel que max(|kA|,|A(k)|)>Nb. Ceci répond affirmativement à des questions posées dans Erdős et Szemerédi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) et, récemment, par S. Konjagin (communication privée). La méthode est basée sur des arguments d'analyse harmonique dans l'esprit de Chang (Ann. Math. 157 (2003) 939–957) et de la combinatoire sur des graphes.
Let be a finite set of integers of cardinality |A|=N⩾2. Given a positive integer k, denote kA (resp. A(k)) the set of all sums (resp. products) of k elements of A. We prove that for all b>1, there exists k=k(b) such that max(|kA|,|A(k)|)>Nb. This answers affirmably questions raised in Erdős and Szemerédi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) and recently, by S. Konjagin (private communication). The method is based on harmonic analysis techniques in the spirit of Chang (Ann. Math. 157 (2003) 939–957) and combinatorics on graphs.
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@article{CRMATH_2003__337_8_499_0,
author = {Bourgain, Jean and Chang, Mei-Chu},
title = {On multiple sum and product sets of finite sets of integers},
journal = {Comptes Rendus. Math\'ematique},
pages = {499--503},
publisher = {Elsevier},
volume = {337},
number = {8},
year = {2003},
doi = {10.1016/j.crma.2003.08.010},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2003.08.010/}
}
TY - JOUR AU - Bourgain, Jean AU - Chang, Mei-Chu TI - On multiple sum and product sets of finite sets of integers JO - Comptes Rendus. Mathématique PY - 2003 SP - 499 EP - 503 VL - 337 IS - 8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2003.08.010/ DO - 10.1016/j.crma.2003.08.010 LA - en ID - CRMATH_2003__337_8_499_0 ER -
%0 Journal Article %A Bourgain, Jean %A Chang, Mei-Chu %T On multiple sum and product sets of finite sets of integers %J Comptes Rendus. Mathématique %D 2003 %P 499-503 %V 337 %N 8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2003.08.010/ %R 10.1016/j.crma.2003.08.010 %G en %F CRMATH_2003__337_8_499_0
Bourgain, Jean; Chang, Mei-Chu. On multiple sum and product sets of finite sets of integers. Comptes Rendus. Mathématique, Tome 337 (2003) no. 8, pp. 499-503. doi : 10.1016/j.crma.2003.08.010. http://www.numdam.org/articles/10.1016/j.crma.2003.08.010/
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