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

Claire Burrin; Matthias Gröbner

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arXiv · 2502.17678

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

Claire Burrin·Matthias Gröbner

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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.

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Claire Burrin, Matthias Gröbner. 2025-02-24. Equidistribution, covering radius, and Diophantine approximation for rational points on the sphere. https://arxiv.org/abs/2502.17678

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