{
  "id": "1704.01317",
  "version": "v1",
  "published": "2017-04-05T09:19:38.000Z",
  "updated": "2017-04-05T09:19:38.000Z",
  "title": "The exceptional sets on the run-length function of beta-expansions",
  "authors": [
    "Lixuan Zheng",
    "Min Wu",
    "Bing Li"
  ],
  "comment": "18 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Let $\\beta > 1$ and the run-length function $r_n(x,\\beta)$ be the maximal length of consecutive zeros amongst the first n digits in the $\\beta$-expansion of $x\\in[0,1]$. The exceptional set $$E_{\\max}^{\\varphi}=\\left\\{x \\in [0,1]:\\liminf_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{\\varphi(n)}=0, \\limsup_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{\\varphi(n)}=+\\infty\\right\\}$$ is investigated, where $\\varphi: \\mathbb{N} \\rightarrow \\mathbb{R}^+$ is a monotonically increasing function with $\\lim\\limits_{n\\rightarrow \\infty }\\varphi(n)=+\\infty$. We prove that the set $E_{\\max}^{\\varphi}$ is either empty or of full Hausdorff dimension and residual in $[0,1]$ according to the increasing rate of $\\varphi$ .",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-05T09:19:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "28A80"
    ],
    "keywords": [
      "exceptional set",
      "run-length function",
      "beta-expansions",
      "full hausdorff dimension",
      "maximal length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}