Number Theory Seminar

Speaker: Michael Björklund (Chalmers)

Title: On the long-range dependence (hyperuniformity) in quasi-crystalline point processes

Abstract: A quasi-crystallline point process is a translation-invariant probability measure on the space of uniformly discrete configurations in a Euclidean space whose difference sets are uniformly discrete approximate subgroups. Prominent examples stem from cut-and-projecting higher dimensional periodic point processes, as well as thinned versions thereof. In this talk we discuss the long-range dependence in such point processes, by which we mean the asymptotic behavior of the diffraction measure of shrinking balls, centered around zero (I will explain the terminology). It turns out that the strength of long-range dependence has to do with Diophantine properties of the  quasi-crystalline random point sets, and using very well-approximable numbers, it is possible to construct quasi-crystalline point processes whose second-order behaviors are even super-Poissonian (the nomenclature will be explained).  If time permits, I will also mention some new long range phenomena which appear for quasi-crystalline processes in other commutative spaces, such as Heisenberg groups and symmetric spaces (both real and S-adic).

Based on joint works with Tobias Hartnick (Karlsruhe).