Number Theory
Sieving and expanders
Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 155-159.

Let V be an orbit in Zn of a finitely generated subgroup Λ of GLn(Z) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). 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)=SL2.

Soit V l'orbite dans Zn d'un sous-groupe finiment engendré de GLn(Z) don't l'adhérence dans la topologie de Zariski est suffisament grande (p.e. est isomorphe à SL2). 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)=SL2.

Received:
Published online:
DOI: 10.1016/j.crma.2006.05.023
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
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Bourgain, Jean; Gamburd, Alex; Sarnak, Peter. Sieving and expanders. Comptes Rendus. Mathématique, Volume 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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