Counting rational points close to $p$-adic integers and applications in Diophantine approximation

TL;DR

This paper establishes bounds on the number of rational points approximating $p$-adic integers and applies these bounds to determine the Hausdorff dimension of certain $p$-adic approximation sets, advancing Diophantine approximation theory.

Contribution

It introduces new bounds on rational points near $p$-adic integers and applies lattice counting and pigeonhole techniques to analyze $p$-adic Diophantine approximation sets.

Findings

Established upper and lower bounds on rational approximations

Determined Hausdorff dimension of $p$-adic approximation sets

Constructed local ubiquitous systems for approximation points

Abstract

We find upper and lower bounds on the number of rational points that are $\psi$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of $p$-adic weighted simultaneously approximable points intersected with $p$-adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that $\tau$-approximate a $p$-adic integer form a set of resonant points that can be used to construct a local ubiquitous system of rectangles.