{
  "id": "2012.13299",
  "version": "v1",
  "published": "2020-12-24T15:52:32.000Z",
  "updated": "2020-12-24T15:52:32.000Z",
  "title": "Classification and statistics of cut and project sets",
  "authors": [
    "René Rühr",
    "Yotam Smilansky",
    "Barak Weiss"
  ],
  "categories": [
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-24T15:52:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "classification",
      "statistics",
      "define ratner-marklof-strombergsson measures",
      "siegel summation formula",
      "error estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}