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Comptes Rendus Mathématique
Volume 343, n° 3
pages 155-159 (août 2006)
Doi : 10.1016/j.crma.2006.05.023
Received : 29 May 2006 ; 
Sieving and expanders
Cribles et expanseurs
 

Jean Bourgain a , Alex Gamburd a, b , Peter Sarnak a, c
a School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA 
b Department of Mathematics, University of California, Santa Cruz, USA 
c Department of Mathematics, Princeton University, USA 

Abstract

Let V be an orbit in   of a finitely generated subgroup of   whose Zariski closure   is suitably large (e.g. isomorphic to  ). 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  . To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The full text of this article is available in PDF format.
Résumé

Soit V lʼorbite dans   dʼun sous-groupe finiment engendré de   donʼt lʼadhérence dans la topologie de Zariski est suffisament grande (p.e. est isomorphe à  ). Nous developpons une crible combinatoire de Brun a fin dʼestimer le nombre de points de V pour lesquels un system de polynômes donnés prennent des valeurs premières ou presque premières. Des propriétés dʼexpansion de certain « graphes de congruence » y jouent un rôle crucial, quʼon établi dans le cas  . Pour citer cet article : J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The full text of this article is available in PDF format.
  The first author was supported in part by NSF grant DMS-0322370. The second author was supported in part by NSF grant DMS-0111298 and DMS-0501245. The third author was supported in part by Oscar Veblen Fund (IAS) and the NSF.


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