Acta Mathematica

Prime and almost prime integral points on principal homogeneous spaces

Amos Nevo and Peter Sarnak

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Abstract

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.

Note

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.

Article information

Source
Acta Math., Volume 205, Number 2 (2010), 361-402.

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

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892503

Digital Object Identifier
doi:10.1007/s11511-010-0057-4

Mathematical Reviews number (MathSciNet)
MR2746350

Zentralblatt MATH identifier
1233.11102

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

Rights
2010 © Institut Mittag-Leffler

Citation

Nevo, Amos; Sarnak, Peter. Prime and almost prime integral points on principal homogeneous spaces. Acta Math. 205 (2010), no. 2, 361--402. doi:10.1007/s11511-010-0057-4. https://projecteuclid.org/euclid.acta/1485892503


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