Analysis, Spectra and Number Theory, a Conference in Honor of Peter Sarnak’s 61st Birthday

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 the lecture I will explore what happens beyond uniform distribution, giving evidence to randomness on smaller scales, examining quantities such as the electrostatic potential and nearest neighbour statistics. (joint work with J. Bourgain and P. Sarnak).