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Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

Humphries, Peter and Radziwiłł, Maksym (2019) Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20210825-184526761

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Abstract

We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a "microsquare", a conjecture of Bourgain and Rudnick on the number of lattice points lying in small balls on the surface of the sphere, the covering radius of the sphere, and the distribution of lattice points in almost all thin regions lying on the surface of the sphere. Finally, we show that for a density 1 subsequence of squarefree integers, the variance exhibits a different asymptotic behaviour for balls of volume (log n)^(−δ )with 0< δ < 1/16. We also obtain analogous results for Heegner points and closed geodesics. Interestingly, we are able to prove some slightly stronger results for closed geodesics than for Heegner points or lattice points on the surface of the sphere. A crucial observation that underpins our proof is the different behaviour of weights functions for annuli and for balls.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1910.01360arXivDiscussion Paper
Additional Information:The first author is supported by the European Research Council grant agreement 670239. The second author acknowledges support of NSF grant DMS-1902063 and of a Sloan fellowship.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)670239
NSFDMS-1902063
Alfred P. Sloan FoundationUNSPECIFIED
Classification Code:2010 Mathematics Subject Classification. 11E16 (primary); 11F67 (secondary)
Record Number:CaltechAUTHORS:20210825-184526761
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210825-184526761
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110524
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:26 Aug 2021 21:26
Last Modified:26 Aug 2021 21:26

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