Lifting, restricting and sifting integral points on affine homogeneous varieties

TL;DR

This paper advances the understanding of counting and bounding integral points on affine homogeneous varieties using ergodic theory and spectral gap techniques, providing effective bounds and solutions to congruence problems.

Contribution

It extends previous methods to establish effective bounds on lifting, integral points, and almost prime solutions in affine homogeneous varieties.

Findings

Effective upper bounds on lifting solutions of congruences

Bounds on the number of integral points on subvarieties

Lower bounds on almost prime points

Abstract

In a previous paper {GN2} an effective solution of the lattice point counting problem in general domains in semisimple S-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action of G on G/Gamma, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in {NS} and use them to establish several useful consequences of this property, including : Effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties, effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties, effective lower bounds on the number of almost prime points on symmetric varieties, and effective upper bounds on almost prime solutions of Linnik-type congruence problems in homogeneous varieties.