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2010 Prime and almost prime integral points on principal homogeneous spaces
Amos Nevo, Peter Sarnak
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Acta Math. 205(2): 361-402 (2010). DOI: 10.1007/s11511-010-0057-4


We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.

Funding Statement

A. N. was supported by the Institute for Advanced Study, Princeton, and ISF grant 975/05. P. S. was supported by an NSF grant and BSF grant 2006254.


Download Citation

Amos Nevo. Peter Sarnak. "Prime and almost prime integral points on principal homogeneous spaces." Acta Math. 205 (2) 361 - 402, 2010.


Received: 6 November 2008; Revised: 5 October 2009; Published: 2010
First available in Project Euclid: 31 January 2017

zbMATH: 1233.11102
MathSciNet: MR2746350
Digital Object Identifier: 10.1007/s11511-010-0057-4

Keywords: affine sieve , arithmetic lattices , lattice points , mean ergodic theorem , prime numbers , principal homogeneous spaces , semisimple groups , spectral gap

Rights: 2010 © Institut Mittag-Leffler

Vol.205 • No. 2 • 2010
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