[Un théorème somme-produit et des estimées des sommes exponentielles dans les classes de résidus avec module composé comportant un nombre borné de nombres premiers.]
Nous présentons dans cette Note une extension des résultats obtenus par Bourgain, Konyagin et Glibichuk pour les modules composés q dont la factorization ne comporte qu'un nombre borné de nombres premiers ‘grands’. D'abord nous démontrons un théorème « somme-produit » pour les sous-ensembles A de , affirmant que si et n'a pas de « grosse » intersection avec une translatée d'un sous-anneau de . Ensuite on obtient des estimées sur des sommes exponentielles, en particulier associées à des sous-groupes multiplicatifs . Ils s'appliquent aux sommes de type Heilbronn pour lesquelles on établit des estimées non-trivials.
In this Note, we extend the results of Bourgain, Konyagin and Glibichuk to certain composite moduli q involving few ‘large’ primes. First a ‘sum-product’ theorem for subsets A of is obtained, ensuring that provided and A does not have a ‘large’ intersection with a translate of a subring. Next, exponential sum estimates are established. In particular nontrivial bounds are obtained for the exponential sums associated to a multiplicative subgroup , with applications to Heilbronn-type sums.
Accepté le :
Publié le :
Jean Bourgain 1 ; Mei-Chu Chang 2
@article{CRMATH_2004__339_7_463_0, author = {Jean Bourgain and Mei-Chu Chang}, title = {Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors}, journal = {Comptes Rendus. Math\'ematique}, pages = {463--466}, publisher = {Elsevier}, volume = {339}, number = {7}, year = {2004}, doi = {10.1016/j.crma.2004.08.007}, language = {en}, }
TY - JOUR AU - Jean Bourgain AU - Mei-Chu Chang TI - Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors JO - Comptes Rendus. Mathématique PY - 2004 SP - 463 EP - 466 VL - 339 IS - 7 PB - Elsevier DO - 10.1016/j.crma.2004.08.007 LA - en ID - CRMATH_2004__339_7_463_0 ER -
%0 Journal Article %A Jean Bourgain %A Mei-Chu Chang %T Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors %J Comptes Rendus. Mathématique %D 2004 %P 463-466 %V 339 %N 7 %I Elsevier %R 10.1016/j.crma.2004.08.007 %G en %F CRMATH_2004__339_7_463_0
Jean Bourgain; Mei-Chu Chang. Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors. Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 463-466. doi : 10.1016/j.crma.2004.08.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.007/
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