{
  "id": "1603.08402",
  "version": "v1",
  "published": "2016-03-28T15:33:52.000Z",
  "updated": "2016-03-28T15:33:52.000Z",
  "title": "Approximation orders of real numbers by $β$-expansions",
  "authors": [
    "Lulu Fang",
    "Min Wu",
    "Bing Li"
  ],
  "comment": "35 pages. Any comments are welcome. Thanks!",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $\\beta$-expansions with the exponential order $\\beta^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. At last, some results related to the orbits of real numbers under $\\beta$-transformation, the shrinking target type problem for $\\beta$-transformations and the Diophantine-type problem for $\\beta$-expansions are also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-28T15:33:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "37B10",
      "28A80"
    ],
    "keywords": [
      "real numbers",
      "approximation orders",
      "expansions",
      "shrinking target type problem",
      "diophantine-type problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}