A quantitative Khintchine–Groshev theorem for S-arithmetic diophantine approximation

Abstract

In Schmidt (1960), 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 (Alam et al., 2021).

In this paper, we generalize this new approach to $S$-arithmetic spaces and obtain a quantitative version of an $S$-arithmetic Khintchine–Groshev theorem. During the process, we consider a new, but still natural $S$-arithmetic analog of Diophantine approximation, which is different from the one formerly established (see Kleinbock and Tomanov, 2007). Hence for the sake of completeness, we also deal with the convergent case of the Khintchine–Groshev theorem, based on this new generalization.