{
  "id": "1609.08724",
  "version": "v1",
  "published": "2016-09-28T01:49:54.000Z",
  "updated": "2016-09-28T01:49:54.000Z",
  "title": "Hausdorff dimension of the set in irrational rotations",
  "authors": [
    "Dong Han Kim",
    "Michał Rams",
    "Baowei Wang"
  ],
  "comment": "15pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $\\theta$ be an irrational number and $\\varphi: {\\mathbb N} \\to {\\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_\\varphi(\\theta) =\\Big\\{y \\in \\mathbb R: \\|n\\theta- y\\|<\\varphi(n), \\ {\\text{for infinitely many}}\\ n\\in {\\mathbb N} \\Big\\}, $$ i.e. the set of points which are approximated by the irrational rotation with respect to the error function $\\varphi(n)$. In this article, we give a complete description of the Hausdorff dimension of $E_\\varphi(\\theta)$ for any monotone function $\\varphi$ and any irrational $\\theta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-28T01:49:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "11J71",
      "28A80"
    ],
    "keywords": [
      "hausdorff dimension",
      "irrational rotation",
      "error function",
      "monotone function",
      "irrational number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}