{
  "id": "1603.01081",
  "version": "v1",
  "published": "2016-03-03T13:00:08.000Z",
  "updated": "2016-03-03T13:00:08.000Z",
  "title": "Beta-expansion and continued fraction expansion of real numbers",
  "authors": [
    "Lulu Fang",
    "Min Wu",
    "Bing Li"
  ],
  "comment": "16 pages. Any comments are welcome. Thank you very much! arXiv admin note: text overlap with arXiv:1601.02202",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "Let $\\beta > 1$ be a real number and $x \\in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\\beta$-expansion of $x$ ($n \\in \\mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\\log2\\log\\beta)/\\pi^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \\in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\\log2\\log\\beta)/\\pi^2$ decays to 0 exponentially as $n$ tends to $\\infty$, which generalizes the result of Faivre \\cite{lesFai97} from $\\beta = 10$ to any $\\beta >1$. Moreover, we also discuss which of the $\\beta$-expansion and continued fraction expansion yields the better approximations of real numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-03T13:00:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "11K50",
      "60F15"
    ],
    "keywords": [
      "real number",
      "beta-expansion",
      "lebesgue measure",
      "continued fraction expansion yields",
      "exact number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}