2024/02/20 by Christopher Lutsko, Lutsko, Christopher
Mathematics · #Analytic Number Theory Research #Mathematical Approximation and Integration #Point processes and geometric inequalities #math.NT
paper · pdf · doi:10.48550/arxiv.2402.12822
openalex publication_date 2024/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Let \widehat\mathcal E(n) denote the set of integer points on the sphere |x|2=n, projected radially onto the unit sphere. Under the usual congruence conditions on n, Duke proved that these points become equidistributed as n→∞. To study their finer-scale distribution, we consider the variance of the number of projected lattice points contained in a spherical cap. Bourgain, Rudnick, and Sarnak conjectured an asymptotic formula for this variance. We prove an unconditional upper bound of the conjectured order of magnitude after averaging over the squared radius n, and we obtain a corresponding estimate for averages over sufficiently long intervals.