Classification and statistics of cut-and-project sets

René Rühr, Yotam Smilansky, Barak Weiss

Research output: Contribution to journalArticlepeer-review

Abstract

We define Ratner–Marklof–Strömbergsson measures (following Marklof and Strömbergsson (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.

Original languageEnglish
Pages (from-to)3575-3638
Number of pages64
JournalJournal of the European Mathematical Society
Volume26
Issue number9
DOIs
StatePublished - 2024

Keywords

  • cut and project sets
  • homogeneous flows
  • Quasicrystals
  • statistics

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics

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