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

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.