{
  "id": "1502.05007",
  "version": "v1",
  "published": "2015-02-17T19:10:38.000Z",
  "updated": "2015-02-17T19:10:38.000Z",
  "title": "Resonant sets, diophantine conditions, and Hausdorff measures for translation surfaces",
  "authors": [
    "Luca Marchese",
    "Rodrigo Treviño",
    "Steffen Weil"
  ],
  "comment": "41 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We consider Teichm\\\"uller geodesics in strata of translation surfaces. We prove a Jarn\\'ik-type inequality for geodesics bounded in some compact part of the stratum and we establish generalized logarithmic laws for geodesics admitting excursions to infinity at a given prescribed rate. Our main tool are planar resonant sets arising from a given translation surface, that is the countable sets of directions of its saddle connections or of its closed geodesics, filtered according to length. We study approximations of a general direction by the elements of a resonant set. In an abstract setting, and assuming specific metric properties for resonant sets, we prove a dichotomy for the Hausdorff measure of well approximable directions and an estimate on the dimension of badly approximable directions. Then we prove that resonant sets arising from a translation surface satisfies the required metric properties. Our techniques also give estimates on the Hausdorff dimension of the set of directions in a rational billiard having fast recurrence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-17T19:10:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff measure",
      "diophantine conditions",
      "resonant sets arising",
      "assuming specific metric properties",
      "planar resonant sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}