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 pointsComment: 2 figures. Included reviewer comments. Accepted for the Contemporary
Mathematics proceedings of Constructive Functions 201
