$p$-adic Directions of Primitive Vectors

TL;DR

This paper proves the uniform distribution of primitive integer vectors' projections on the p-adic unit sphere as their real norm increases, using lattice point counting in S-arithmetic groups.

Contribution

It establishes the distribution of primitive vectors in the p-adic setting, extending classical distribution results to a new arithmetic context.

Findings

Primitive vectors become uniformly distributed in the p-adic unit sphere

Distribution results hold as the real norm tends to infinity

Method involves counting lattice points in semi-simple S-arithmetic groups

Abstract

Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform distribution of the projections of primitive $\mathbb{Z}^{2}$ points in the $p$-adic unit sphere, as their (real) norm tends to infinity. The proof is via counting lattice points in semi-simple $S$-arithmetic groups.