Classification and statistics of cut and project sets

TL;DR

This paper introduces a classification of measures on cut-and-project sets in higher-dimensional spaces, linking homogeneous dynamics with algebraic groups, and derives asymptotic counting results with error estimates.

Contribution

It classifies invariant measures on cut-and-project sets using algebraic and homogeneous dynamics, extending classical summation formulas and counting results.

Findings

Classified Ratner-Marklof-Strombergsson measures on cut-and-project sets.

Derived asymptotic formulas with error estimates for point and patch counting.

Extended classical results like Siegel's formula to higher-dimensional cut-and-project sets.

Abstract

We define Ratner-Marklof-Strombergsson measures. These are probability measures supported on cut-and-project sets in R^d (d > 1) which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(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.