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Local Statistics of Lattice Points on the Sphere

Author(s): Bourgain, Jean; Sarnak, Peter C; Rudnick, Zeev

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Abstract: A celebrated result of Legendre and Gauss determines which integers can be represented as a sum of three squares, and for those it is typically the case that there are many ways of doing so. These different representations give collections of points on the unit sphere, and a fundamental result, conjectured by Linnik, is that under a simple condition these become uniformly distributed on the sphere. In this note we survey some of our recent work, which explores what happens beyond uniform distribution, giving evidence to randomness on smaller scales. We treat the electrostatic energy, local statistics such as the point pair statistic (Ripley’s function), nearest neighbour statistics, minimum spacing and covering radius. We briefly discuss the situation in other dimensions, which is very different. In an appendix we compute the corresponding quantities for random points.
Publication Date: 2016
Electronic Publication Date: 2016
Citation: Bourgain, Jean, Sarnak, Peter, Rudnick, Zeev. (2016). Local Statistics of Lattice Points on the Sphere. MODERN TRENDS IN CONSTRUCTIVE FUNCTION THEORY, 661 (269 - 282. doi:10.1090/conm/661/13287
DOI: doi:10.1090/conm/661/13287
ISSN: 0271-4132
Pages: 269 - 282
Type of Material: Conference Article
Version: Author's manuscript

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