Prime and almost prime integral points on principal homogeneous spaces

TL;DR

This paper extends the affine sieve method to orbits of congruence subgroups acting on affine space, providing effective bounds for prime factorization properties of polynomial values and establishing prime-rich matrices density.

Contribution

It introduces effective bounds for saturation numbers in affine sieves on group orbits, including sharp results for matrices with prime entries, advancing understanding of prime distribution in algebraic structures.

Findings

Effective bounds for saturation numbers in affine sieves.

Sharp results for matrices with prime entries at fixed determinant.

Establishment of Zariski density of prime-entry matrices.

Abstract

We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple 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 and the orbit is 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.