TY - CHAP

T1 - Local statistics of lattice points on the sphere

AU - Bourgain, Jean

AU - Sarnak, Peter

AU - Rudnick, Zeév

N1 - Publisher Copyright:
© 2016 American Mathematical Society.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85106855117&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85106855117&partnerID=8YFLogxK

U2 - 10.1090/conm/661/13287

DO - 10.1090/conm/661/13287

M3 - Chapter

AN - SCOPUS:85106855117

T3 - Contemporary Mathematics

SP - 269

EP - 282

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -