Classification and statistics of cut-and-project sets

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