December 10, 2025Open Access

Dual p -adic Diophantine approximation on manifolds

Key Points

The homogeneous p-adic convergence was established for hypersurfaces of dimension at least three with regularity conditions.
The results extend findings by Beresnevich-Dickinson-Velani and Badziahin-Beresnevich-Velani on manifold approximation.
We explored both homogeneous and inhomogeneous settings for p-adic approximation outside monotonic constraints.
Implications highlight the need for continued investigation into convergence aspects of the Generalised Baker–Schmidt Problem.

Abstract

Abstract The Generalised Baker–Schmidt Problem (1970) concerns the Hausdorff measure of the set of -approximable points on a non-degenerate 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 non-degenerate 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 question. 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.

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