Prime and almost prime integral points on principal homogeneous spaces

Amos Nevo, Peter Sarnak

Research output: Contribution to journalArticle

16 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)361-402
Number of pages42
JournalActa Mathematica
Volume205
Issue number2
DOIs
StatePublished - Dec 1 2010

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

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

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