Classification and statistics of cut-and-project sets
Rene Rühr
Technion – Israel Institute of Technology, Haifa, IsraelYotam Smilansky
Rutgers University, Piscataway, USABarak Weiss
Tel Aviv University, Israel
Abstract
We define Ratner–Marklof–Strömbergsson measures (following Marklof and Strömbergsson (2014)). These are probability measures supported on cut-and-project sets in which are invariant and ergodic for the action of the groups or . 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.
Cite this article
Rene Rühr, Yotam Smilansky, Barak Weiss, Classification and statistics of cut-and-project sets. J. Eur. Math. Soc. (2023), published online first
DOI 10.4171/JEMS/1338