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We find upper and lower bounds on the number of rational points that are$\psi$-approximations of some $n$-dimensional $p$-adic integer. Lattice pointcounting techniques are used to find the upper bound result, and a Pigeon-holeprinciple style argument is used to find the lower bound result. We use theseresults to find the Hausdorff dimension for the set of $p$-adic weightedsimultaneously approximable points intersected with $p$-adic coordinatehyperplanes. For the lower bound result we show that the set of rational pointsthat $\tau$-approximate a $p$-adic integer form a set of resonant points thatcan be used to construct a local ubiquitous system of rectangles.

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