Simultaneous Diophantine Approximation in Non-degenerate p-adic Manifolds

TL;DR

This paper establishes a Khintchine-type theorem for S-arithmetic Diophantine approximation on non-degenerate p-adic manifolds, covering both convergence and divergence cases, advancing understanding in p-adic number theory.

Contribution

It proves the convergence case and the divergence part of the S-arithmetic Khintchine-type theorem for non-degenerate p-adic manifolds, extending previous results in the field.

Findings

Proved the convergence case of the theorem.

Established the divergence case for p-adic manifolds.

Extended Diophantine approximation results to S-arithmetic p-adic settings.

Abstract

S-arithmetic Khintchine-type theorem for products of non-degenerate analytic p-adic manifolds is proved for the convergence case. In the p-adic case the divergence part is also obtained.