{
  "id": "1405.4747",
  "version": "v2",
  "published": "2014-05-19T14:37:19.000Z",
  "updated": "2015-10-28T16:38:47.000Z",
  "title": "Subexponentially increasing sum of partial quotients in continued fraction expansions",
  "authors": [
    "Lingmin Liao",
    "Michal Rams"
  ],
  "comment": "11 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate from multifractal analysis point of view the increasing rate of the sum of partial quotients $S\\_n(x)=\\sum\\_{j=1}^n a\\_j(x)$, where $x=[a\\_1(x), a\\_2(x), \\cdots ]$ is the continued fraction expansion of an irrational $x\\in (0,1)$. Precisely, for an increasing function $\\varphi: \\mathbb{N} \\rightarrow \\mathbb{N}$, one is interested in the Hausdorff dimension of the sets \\[ E\\_\\varphi = \\left\\{x\\in (0,1): \\lim\\_{n\\to\\infty} \\frac {S\\_n(x)} {\\varphi(n)} =1\\right\\}. \\] Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case $\\exp(n^\\beta), \\ \\beta \\in [1/2, 1)$. We show that when $\\beta \\in [1/2, 1)$, $E\\_\\varphi$ has Hausdorff dimension $1/2$. Thus surprisingly the dimension has a jump from $1$ to $1/2$ at the increasing rate $\\exp(n^{1/2})$. In a similar way, the distribution of the largest partial quotients is also studied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-19T14:37:19.000Z",
      "abstract": "We investigate from multifractal analysis point of view the increasing rate of the sum of partial quotients $S_n(x)=\\sum_{j=1}^n a_j(x)$, where $x=[a_1(x), a_2(x), \\cdots ]$ is the continued fraction expansion of an irrational $x\\in (0,1)$. Precisely, for an increasing function $\\varphi: \\mathbb{N} \\rightarrow \\mathbb{N}$, one is interested in the Hausdorff dimension of the sets \\[ E_\\varphi = \\left\\{x\\in (0,1): \\lim_{n\\to\\infty} \\frac {S_n(x)} {\\varphi(n)} =1\\right\\}. \\] Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case $\\exp(n^\\beta), \\ \\beta \\in [1/2, 1)$. We show that when $\\beta \\in [1/2, 1)$, $E_\\varphi$ has Hausdorff dimension $1/2$. Thus surprisingly the dimension has a jump from $1$ to $1/2$ at the increasing rate $\\exp(n^{1/2})$. In a similar way, the distribution of the largest partial quotients is also studied.",
      "comment": "10 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-10-28T16:38:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continued fraction expansion",
      "subexponentially increasing sum",
      "hausdorff dimension",
      "multifractal analysis point",
      "increasing rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.4747L"
    }
  }
}