@article{abc4246cc54840dfb64accf41e40e1c5,
title = "Classification and statistics of cut-and-project sets",
abstract = "We define Ratner–Marklof–Str{\"o}mbergsson measures (following Marklof and Str{\"o}mbergsson (2014)). These are probability measures supported on cut-and-project sets in Rd .d ≥ 2/ which are invariant and ergodic for the action of the groups ASLd .R/ or SLd .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.",
keywords = "Quasicrystals, cut and project sets, homogeneous flows, statistics",
author = "Ren{\'e} R{\"u}hr and Yotam Smilansky and Barak Weiss",
note = "Publisher Copyright: {\textcopyright} 2023 European Mathematical Society.",
year = "2024",
doi = "10.4171/JEMS/1338",
language = "אנגלית",
volume = "26",
pages = "3575--3638",
journal = "Journal of the European Mathematical Society",
issn = "1435-9855",
publisher = "European Mathematical Society Publishing House",
number = "9",
}