Dual $p$-adic Diophantine approximation on manifolds

TL;DR

This paper advances the understanding of $p$-adic Diophantine approximation on manifolds by proving convergence results for hypersurfaces and other classes, extending previous divergence results.

Contribution

It establishes the first homogeneous $p$-adic convergence theorems for hypersurfaces of dimension at least three, with some regularity conditions, and extends to inhomogeneous cases.

Findings

Proved homogeneous $p$-adic convergence for hypersurfaces of dimension ≥3.

Extended convergence results to certain other manifolds.

Did not restrict to monotonic approximation functions.

Abstract

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The divergence part has also been resolved for the $p$-adic setting by Datta-Ghosh in 2022 for the inhomogeneous setting. The corresponding convergence counterpart represents a challenging open problem. In this paper, we prove the homogeneous $p$-adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.