Quantitative Diophantine approximation with congruence conditions

TL;DR

This paper extends the Khintchine-Groshev theorem to include congruence conditions, providing a quantitative measure of approximation in Diophantine problems using lattice point counting techniques.

Contribution

It introduces a quantitative version of the Khintchine-Groshev theorem incorporating congruence conditions, utilizing lattice counting and variance bounds.

Findings

Established a quantitative Khintchine-Groshev theorem with congruence conditions

Applied Schmidt's lattice point counting method to Diophantine approximation

Provided variance bounds on the space of lattices for the proof

Abstract

In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.