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

2021/02/17 by Benjamin Ward, Ward, Benjamin
#11J61 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2102.09070

Abstract

We find upper and lower bounds on the number of rational points that are ψ-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 τ-approximate a p-adic integer form a set of resonant points that can be used to construct a local ubiquitous system of rectangles.

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