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arXiv · 2102.09070

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

Benjamin Ward
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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.

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Benjamin Ward. 2021-02-17. Counting rational points close to $p$-adic integers and applications in Diophantine approximation. https://arxiv.org/abs/2102.09070

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