Pär Kurlberg: Poisson spacings for lattice points on circles

Abstract

We will investigate the distribution of \(\mathbb{Z}^2\)-lattice points lying on circles. Along a density one subsequence the angles of lattice points on circles are known to be uniformly distributed as the radius tends to infinity; in fact the angles are "very well distributed" in the sense of the discrepancy being *lower* than that of a random collection of points. A refined question is how lattice points are spaced at the local scale, i.e., when rescaled so that the mean spacing is one. I will discuss recent joint work with Steve Lester in which we show that the local spacing statistics are Poissonian along a density one subsequence of admissible radii.