{
  "id": "1404.1564",
  "version": "v2",
  "published": "2014-04-06T11:09:54.000Z",
  "updated": "2014-08-08T16:14:15.000Z",
  "title": "Visibility and directions in quasicrystals",
  "authors": [
    "Jens Marklof",
    "Andreas Strömbergsson"
  ],
  "comment": "20 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MG",
    "math.MP",
    "math.NT"
  ],
  "abstract": "It is well known that a positive proportion of all points in a $d$-dimensional lattice is visible from the origin, and that these visible lattice points have constant density in $\\mathbb{R}^d$. In the present paper we prove an analogous result for a large class of quasicrystals, including the vertex set of a Penrose tiling. We furthermore establish that the statistical properties of the directions of visible points are described by certain $\\operatorname{SL}(d,\\mathbb{R})$-invariant point processes. Our results imply in particular existence and continuity of the gap distribution for directions in certain two-dimensional cut-and-project sets. This answers some of the questions raised by Baake et al. in [arXiv:1402.2818].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-08T16:14:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "directions",
      "quasicrystals",
      "visibility",
      "two-dimensional cut-and-project sets",
      "invariant point processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.1564M"
    }
  }
}