A quantitative Khintchine-Groshev theorem for S-arithmetic Diophantine approximation

TL;DR

This paper extends a quantitative Khintchine-Groshev theorem to S-arithmetic spaces, providing new insights into Diophantine approximation in these settings and including convergence cases.

Contribution

It generalizes recent approaches to S-arithmetic Diophantine approximation, introducing a new analog and relaxing previous constraints.

Findings

Established a quantitative S-arithmetic Khintchine-Groshev theorem

Introduced a new S-arithmetic Diophantine approximation framework

Addressed both divergence and convergence cases

Abstract

In his 1960 paper, Schmidt studied a quantitative type of Khintchine-Groshev theorem for general (higher) dimensions. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more generally, to add on the congruence condition by M. Alam, A. Ghosh, and S. Yu. In this paper, we generalize this new approach to S-arithmetic spaces and obtain a quantitative version of an S-arithmetic Khintchine-Groshev theorem. In fact, we consider a new S-arithmetic analog of Diophantine approximation, which is different from the one formerly established (see the 2007 paper of D. Kleinbock and G. Tomanov). Hence for the sake of completeness, we also deal with the convergence case of the Khintchine-Groshev theorem, based on this new generalization.