Sieving and expanders⁎⁎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.

Abstract

Let V be an orbit in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> of a finitely generated subgroup Λ of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> whose Zariski closure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Zcl</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> is suitably large (e.g. isomorphic to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi mathvariant="normal">SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> ). 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Zcl</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi mathvariant="normal">SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> .