Sieving and expanders

Bourgain, Jean; Gamburd, Alex; Sarnak, Peter

doi:10.1016/j.crma.2006.05.023

Number Theory

Sieving and expanders
[Cribles et expanseurs]

Présenté par : Jean Bourgain

Bourgain, Jean ¹ ; Gamburd, Alex ^1,² ; Sarnak, Peter ^1,³

¹ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
² Department of Mathematics, University of California, Santa Cruz, USA
³ Department of Mathematics, Princeton University, USA

Comptes Rendus. Mathématique, Tome 343 (2006) no. 3, pp. 155-159.

Résumé
Abstract

Soit V l'orbite dans $Z^{n}$ d'un sous-groupe finiment engendré de ${GL}_{n} (Z)$ don't l'adhérence dans la topologie de Zariski est suffisament grande (p.e. est isomorphe à ${SL}_{2}$ ). Nous developpons une crible combinatoire de Brun a fin d'estimer le nombre de points de V pour lesquels un system de polynômes donnés prennent des valeurs premières ou presque premières. Des propriétés d'expansion de certain « graphes de congruence » y jouent un rôle crucial, qu'on établi dans le cas $Zcl (Λ) = {SL}_{2}$ .

Let V be an orbit in $Z^{n}$ of a finitely generated subgroup Λ of ${GL}_{n} (Z)$ whose Zariski closure $Zcl (Λ)$ is suitably large (e.g. isomorphic to ${SL}_{2}$ ). We develop a Brun combinatorial sieve for estimating the number of points on V for which a fixed set of integral polynomials take prime or almost prime values. A crucial role is played by the expansion property of the ‘congruence graphs’ that we associate with V. This expansion property is established when $Zcl (Λ) = {SL}_{2}$ .

Reçu le : 2006-05-29
Publié le : 2006-07-12

DOI : 10.1016/j.crma.2006.05.023

Affiliations des auteurs :

Bourgain, Jean ¹ ; Gamburd, Alex ^{1, 2} ; Sarnak, Peter ^{1, 3}

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Bourgain, Jean; Gamburd, Alex; Sarnak, Peter. Sieving and expanders. Comptes Rendus. Mathématique, Tome 343 (2006) no. 3, pp. 155-159. doi : 10.1016/j.crma.2006.05.023. http://www.numdam.org/articles/10.1016/j.crma.2006.05.023/

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