Diophantine aspects of long range dependence in quasi-crystalline point processes

2:00 pm, Wednesday, 21st February

Abstract

The number variance N_R of a point process in d-dimensional Euclidean space is defined as the variance of the number of points in a centered Euclidean ball of radius R. For the most random of all point processes, the Poisson point process, N_R is proportional to the volume of the ball, i.e. N_R grows like R^{d}. On the other hand, for the least random point processes, the periodic ones, N_R grows like R^{d-1} (on average). Following the seminal works of Stillinger and Torquato from the early 2000s, we say that a point process is hyperuniform (or exhibit long range dependence) if the number variance N_R is o(R^d). Besides periodic point processes, determinental point processes (coming from projections) and (certain) quasi-crystalline point processes are hyperuniform. In this talk I will discuss recent works with Tobias Hartnick (Karlsruhe) and Mattias Bylehn (Gothenburg) which explore different number-theoretic and geometric aspects of hyper-uniformity for quasi-crystalline point processes in Euclidean spaces, in Heisenberg geometries and in hyperbolic spaces, and highlight a few surprises on the way. All notions from point process theory will be defined.