{
  "id": "1305.2088",
  "version": "v1",
  "published": "2013-05-09T13:39:20.000Z",
  "updated": "2013-05-09T13:39:20.000Z",
  "title": "Range-Renewal Structure in Continued Fractions",
  "authors": [
    "Jun Wu",
    "Jian-Sheng Xie"
  ],
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "Let $\\omega=[a_1, a_2, \\cdots]$ be the infinite expansion of continued fraction for an irrational number $\\omega \\in (0,1)$; let $R_n (\\omega)$ (resp. $R_{n, \\, k} (\\omega)$, $R_{n, \\, k+} (\\omega)$) be the number of distinct partial quotients each of which appears at least once (resp. exactly $k$ times, at least $k$ times) in the sequence $a_1, \\cdots, a_n$. In this paper it is proved that for Lebesgue almost all $\\omega \\in (0,1)$ and all $k \\geq 1$, $$ \\displaystyle \\lim_{n \\to \\infty} \\frac{R_n (\\omega)}{\\sqrt{n}}=\\sqrt{\\frac{\\pi}{\\log 2}}, \\quad \\lim_{n \\to \\infty} \\frac{R_{n, \\, k} (\\omega)}{R_n (\\omega)}=\\frac{C_{2 k}^k}{(2k-1) \\cdot 4^k}, \\quad \\lim_{n \\to \\infty} \\frac{R_{n, \\, k} (\\omega)}{R_{n, \\, k+} (\\omega)}=\\frac{1}{2k}. $$ The Hausdorff dimensions of certain level sets about $R_n$ are discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-09T13:39:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continued fraction",
      "range-renewal structure",
      "distinct partial quotients",
      "irrational number",
      "infinite expansion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.2088W"
    }
  }
}