  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.
