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

Abstract

In [18], Schmidt studied a quantitative type of Khintchine-Groshev theorem. 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 [1].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 [15]). Hence for the sake of completeness, we also deal with the convergence case of the Khintchine-Groshev theorem, based on this new generalization.