{
  "id": "1804.06608",
  "version": "v1",
  "published": "2018-04-18T09:00:22.000Z",
  "updated": "2018-04-18T09:00:22.000Z",
  "title": "Hausdorff dimensions of sets related to Erdös-Rényi average in beta expansions",
  "authors": [
    "Haibo Chen"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\beta>1$ and $\\phi$ be an integer function defined on $\\mathbb{N}\\setminus\\{0\\}$ satisfying $1\\leq\\phi(n)\\leq n$. Define the level set \\begin{align*} ER_\\phi^\\beta(\\alpha)=\\left\\{x\\in[0,1]\\colon A_\\phi(x,\\beta)=\\alpha\\right\\},\\quad \\alpha\\in I_\\beta, \\end{align*} where $A_\\phi(x,\\beta)$ is the Erd\\\"{o}s-R\\'{e}nyi digit average of $x\\in[0,1]$ associated with the function $\\phi$ in $\\beta$-expansion and $I_\\beta$ is the range of $\\alpha$. In this paper, we will determine the Hausdorff dimension of $ER_\\phi^\\beta(\\alpha)$ under the assumption $\\phi(n)\\to\\infty$ as $n\\to\\infty$ and $\\phi$ is the integer part of a slowly varying sequence. Besides, some generalizations in $\\beta$-expansion to the classic work \\cite{Be} of Besicovitch are also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-18T09:00:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "28A80"
    ],
    "keywords": [
      "hausdorff dimension",
      "beta expansions",
      "erdös-rényi average",
      "digit average",
      "integer function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}