Simultaneous p-adic Diophantine approximation

TL;DR

This paper extends classical Diophantine approximation theorems to the p-adic setting, establishing analogues of key results and analyzing the Hausdorff dimension of approximable points on p-adic manifolds.

Contribution

It develops the theory of weighted p-adic Diophantine approximation, proving analogues of Khintchine, Duffin-Schaeffer, and Jarník-Besicovitch theorems, and provides dimension bounds for approximable points on manifolds.

Findings

Established p-adic analogues of classical approximation theorems.

Derived lower bounds for Hausdorff dimension on p-adic manifolds.

Extended the Mass Transference Principle to the p-adic context.

Abstract

The goal of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to $p$-adic numbers. Firstly, we establish complete analogues of Khintchine's theorem, the Duffin-Schaeffer theorem and the Jarn\'ik-Besicovitch theorem for `weighted' simultaneous Diophantine approximation in the $p$-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on $p$-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu, and a Zero-One law for weighted $p$-adic approximations established in this paper.