2025/02/24 by Claire Burrin, Burrin, Claire, Matthias Gröbner +1
Engineering · Mathematics · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Advanced Numerical Analysis Techniques #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2502.17678
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≫ n-1/2+δ, pointwise equidistribution down to R≫ 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.