{
  "id": "1603.03800",
  "version": "v1",
  "published": "2016-03-11T22:00:26.000Z",
  "updated": "2016-03-11T22:00:26.000Z",
  "title": "Diophantine approximation on matrices and Lie groups",
  "authors": [
    "Menny Aka",
    "Emmanuel Breuillard",
    "Lior Rosenzweig",
    "Nicolas de Saxcé"
  ],
  "comment": "52 pages",
  "categories": [
    "math.NT",
    "math.DS",
    "math.GR"
  ],
  "abstract": "We study the general problem of extremality for metric diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over the rational numbers, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest, in particular we manage to determine the diophantine exponent of random subgroups of certain nilpotent Lie groups in terms of representation theoretic data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-11T22:00:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83",
      "11J13",
      "11K60",
      "22E99",
      "22E25"
    ],
    "keywords": [
      "representation theoretic data",
      "metric diophantine approximation",
      "nilpotent lie groups",
      "sure diophantine exponent",
      "zariski closure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160303800A"
    }
  }
}