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

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.

arXiv comment26 pages, results have been improved to the $n$-dimensional case

References in corpus (6)

• Diophantine approximation on planar curves: the convergence theory
• Successive Minima and Lattice Points
• On the spectrum of Diophantine approximation constants
• Simultaneous approximation on affine subspaces
• Mass transference principle from rectangles to rectangles in Diophantine approximation
• On simultaneous rational approximation to a real number and its integral powers, II