Sieving and expanders

Jean Bourgain; Alex Gamburd; Peter Sarnak

doi:10.1016/j.crma.2006.05.023

Number Theory

Sieving and expanders

Presented by: Jean Bourgain

Jean Bourgain ¹; Alex Gamburd ^1,²; Peter Sarnak ^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, Volume 343 (2006) no. 3, pp. 155-159.

Abstract
Résumé

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 $Zcl (Λ) = {SL}_{2}$ .

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 $Zcl (Λ) = {SL}_{2}$ .

Received: 2006-05-29
Published online: 2006-07-12

DOI: 10.1016/j.crma.2006.05.023

Author's affiliations:

Jean Bourgain ¹; Alex Gamburd ^{1, 2}; Peter Sarnak ^{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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     title = {Sieving and expanders},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {155--159},
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     volume = {343},
     number = {3},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.023},
     language = {en},
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Jean Bourgain; Alex Gamburd; Peter Sarnak. Sieving and expanders. Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 155-159. doi : 10.1016/j.crma.2006.05.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.05.023/

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