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

We find upper and lower bounds on the number of rational points with bounded denominators that are contained in a rectangular neighbourhood of some n -dimensional p -adic integer. To find the upper bound, we use lattice point counting techniques on p -adic approximation lattices, and for the corresponding lower-bound statement, a classical pigeonhole principle-style argument is used. We apply this counting result to prove a statement in the setting of weighted simultaneous p -adic Diophantine approximation on coordinate hyperplanes. For the lower-bound Hausdorff dimension result, we construct a local ubiquitous system of rectangles and then apply the recent Mass Transference Principle result of Wang and Wu (Math. Ann., 2021).