Equidistribution, covering radius, and Diophantine approximation for rational points on the sphere

Abstract

We study the distribution of rational points of fixed height on the sphere at shrinking scales. For the two-dimensional sphere, we prove an unconditional variance estimate for primitive square-level Linnik sets, essentially matching the random-model prediction. We obtain almost-everywhere equidistribution in caps down to the optimal scale $R\gg n^{-1/2+\delta }$, pointwise equidistribution down to $R\gg n^{-1/4+o(1)}$, and Wasserstein equidistribution down to the optimal bound. We also derive applications to covering, intrinsic Diophantine approximation, and Linnik's conjecture on sums of two squares and a mini-square.