Classification and statistics of cut-and-project sets

Abstract

We define Ratner–Marklof–Strömbergsson measures (following Marklof and Strömbergsson (2014)). These are probability measures supported on cut-and-project sets in ${\mathbb{R}}^d$ $(d\ge 2)$ which are invariant and ergodic for the action of the groups ${\mathrm{ASL}}_d(\mathbb{R})$ or ${\mathrm{SL}}_d(\mathbb{R})$. We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project sets.