Metric Diophantine approximation with congruence conditions

TL;DR

This paper extends classical Diophantine approximation results to matrices with congruence conditions, using ergodic theory and the Dani correspondence to analyze rational approximations across multiple congruence classes.

Contribution

It proves a version of the Khinchine--Groshev theorem for matrices under congruence constraints, extending ergodic methods to this setting.

Findings

Established a congruence-conditioned Khinchine--Groshev theorem.

Extended Dani correspondence to quotient spaces by congruence subgroups.

Applied multiple ergodic theorems to simultaneous approximation in congruence classes.

Abstract

We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.