{
  "id": "1401.4148",
  "version": "v2",
  "published": "2014-01-16T19:58:30.000Z",
  "updated": "2015-09-25T11:01:03.000Z",
  "title": "Ergodic Theory and Diophantine approximation for translation surfaces and linear forms",
  "authors": [
    "Jayadev Athreya",
    "Andrew Parrish",
    "Jimmy Tseng"
  ],
  "comment": "The revised version gives details of the approximation argument used to deduce Birkhoff genericity on a certain submanifold from Birkhoff genericity on the whole space. Also the title has been slightly changed from the previous version",
  "categories": [
    "math.DS",
    "math.GT",
    "math.NT"
  ],
  "abstract": "We derive results on the distribution of directions of saddle connections on translation surfaces, using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give a simple proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to W. Schmidt~\\cite{SchmidtMetrical}. The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-16T19:58:30.000Z",
      "title": "Ergodic Theory and Diophantine approximation for linear forms and translation surfaces",
      "abstract": "We give a simple proof of a version of a classical theorem in multi-dimensional Diophantine approximation due to W. Schmidt. While our version is weaker, the proof relies only on the Birkhoff ergodic theorem and the Siegel mean value theorem. Our technique also yields results on systems of linear forms and gives us an analogous result in the setting of translation surfaces.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-09-25T11:01:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D40",
      "11P21",
      "11J20",
      "11J13"
    ],
    "keywords": [
      "translation surfaces",
      "linear forms",
      "ergodic theory",
      "siegel mean value theorem",
      "multi-dimensional diophantine approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.4148A"
    }
  }
}