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 language | English |
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Pages (from-to) | 3575-3638 |
Number of pages | 64 |
Journal | Journal of the European Mathematical Society |
Volume | 26 |
Issue number | 9 |
DOIs | |
State | Published - 2024 |
Keywords
- cut and project sets
- homogeneous flows
- Quasicrystals
- statistics
All Science Journal Classification (ASJC) codes
- General Mathematics
- Applied Mathematics