Sam Pattison

Connections between fine-scale statistics and lattices in higher dimensions

Fine-scale statistics investigates statistical properties of sequences of deterministic sets (such as points, lines or varieties) and how these compare to different random models. Classically, this has focused on the properties of points on the real line. For example, we might look at how the first N points of the sequence n^a modulo 1 distribute in a randomly chosen interval of length 1/N, as N tends to infinity. Apart from when a= 1/2, this sequence is believed to distribute among the intervals like a sequence of uniform i.i.d random variables, with the limiting measure of intervals containing k points being Poissonian (equal to 1/ek!). Surprisingly, when a = 1/2 we get an entirely different limiting distribution. This is due to a surprising connection between this sequence and the distribution of affine unimodular lattice points in the plane, discovered by Elkies and McMullen. The limiting distribution can be defined explicitly on this space of lattices, with convergence following from an equidistribution result for this space.
In this talk, we describe recent work in which we generalize the results of Elkies and Mcmullen to higher dimensions.