{
  "id": "1805.04744",
  "version": "v1",
  "published": "2018-05-12T15:58:49.000Z",
  "updated": "2018-05-12T15:58:49.000Z",
  "title": "Level sets of the run-length function of beta-expansions",
  "authors": [
    "Lixuan Zheng"
  ],
  "comment": "25 pages, 6 theorems",
  "categories": [
    "math.DS"
  ],
  "abstract": "For any $\\beta > 1$, denoted by $r_n(x,\\beta)$ the maximal length of consecutive zeros amongst the first $n$ digits of the $\\beta$-expansion of $x\\in[0,1)$. Let $0\\leq a\\leq b\\leq 1$. The Hausdorff dimension of the level set $$E_{a,b}=\\left\\{x \\in [0,1): \\liminf_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{n}=a,\\ \\limsup_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{n}=b\\right\\}$$ is obtained. As a consequence, the set $$E_a=\\left\\{x \\in [0,1):\\liminf_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{n}=a\\right\\}$$ has Hausdorff dimension ${(1-2a)}^2$ when $0\\leq a\\leq\\frac{1}{2}$. We show that the extremely divergent set $E_{0,1}$ which is of zero Hausdorff dimension is, however, residual which means that it is large from the topological viewpoint. The same problems in the parameter space are also examined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-12T15:58:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "28A80"
    ],
    "keywords": [
      "level set",
      "run-length function",
      "beta-expansions",
      "zero hausdorff dimension",
      "maximal length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}