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arXiv · 2012.13299

Classification and statistics of cut and project sets

René Rühr·Yotam Smilansky·Barak Weiss
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Abstract

We define Ratner-Marklof-Strombergsson measures. These are probability measures supported on cut-and-project sets in R^d (d > 1) which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(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.

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René Rühr, Yotam Smilansky, Barak Weiss. 2020-12-24. Classification and statistics of cut and project sets. https://arxiv.org/abs/2012.13299

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