$S$-arithmetic Inhomogeneous Diophantine approximation on manifolds

TL;DR

This paper establishes new $S$-arithmetic inhomogeneous Diophantine approximation theorems on manifolds, extending classical results to a broader $S$-arithmetic setting with novel divergence and convergence results.

Contribution

It proves the divergence case of $S$-arithmetic inhomogeneous Khintchine theorems on manifolds using Hausdorff measures and ubiquitous systems, and confirms the convergence case including the $S$-arithmetic Baker-Sprindzuk conjectures.

Findings

Divergence results are new for multiple primes in the $S$-arithmetic setting.

Convergence case proven using nondivergence estimates and transference principles.

Established in the context of analytic nondegenerate manifolds.

Abstract

We prove $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using ubiquitous systems. For $S$ consisting of more than one prime, finite or infinite, the divergence results are new even in the homogeneous setting. We also prove the convergence case of the theorem, including in particular, the $S$-arithmetic inhomogeneous counterpart of the Baker-Sprind\v{z}uk conjectures using nondivergence estimates for flows on homogeneous spaces in conjunction with the transference principle of Beresnevich and Velani.