Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors

Jean Bourgain; Mei-Chu Chang

doi:10.1016/j.crma.2004.08.007

Number Theory

Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors
[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.]

Présenté par : Jean Bourgain

Jean Bourgain ¹ ; Mei-Chu Chang ²

¹ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
² Department of Mathematics, University of California, Riverside, California, USA

Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 463-466.

Résumé
Abstract

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 $Z_{q}$ , affirmant que $| A + A | + | A . A | > c {| A |}^{1 + ɛ}$ si $| A | < q^{1 - δ}$ et n'a pas de « grosse » intersection avec une translatée d'un sous-anneau de $Z_{q}$ . Ensuite on obtient des estimées sur des sommes exponentielles, en particulier associées à des sous-groupes multiplicatifs $H < Z_{q}^{*}$ . 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 $Z_{q}$ is obtained, ensuring that $| A + A | + | A . A | > c {| A |}^{1 + ɛ}$ provided $| A | < q^{1 - δ}$ 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 $H < Z_{q}^{*}$ , with applications to Heilbronn-type sums.

Reçu le : 2004-08-19
Accepté le : 2004-08-28
Publié le : 2004-09-28

DOI : 10.1016/j.crma.2004.08.007

Affiliations des auteurs :

Jean Bourgain ¹ ; Mei-Chu Chang ²

¹ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
² Department of Mathematics, University of California, Riverside, California, USA

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     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},
}

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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/

Bibliographie
Cité par

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[3] J. Bourgain, A. Glibichuk, S. Konyagin, Estimate for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc., submitted for publication

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