Classification and statistics of cut-and-project sets

  • Rene Rühr

    Technion – Israel Institute of Technology, Haifa, Israel
  • Yotam Smilansky

    Rutgers University, Piscataway, USA
  • Barak Weiss

    Tel Aviv University, Israel
Classification and statistics of cut-and-project sets cover

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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