{
  "id": "1503.00475",
  "version": "v1",
  "published": "2015-03-02T10:37:04.000Z",
  "updated": "2015-03-02T10:37:04.000Z",
  "title": "Hausdorff dimension of univoque sets and Devil's staircase",
  "authors": [
    "Vilmos Komornik",
    "Derong Kong",
    "Wenxia Li"
  ],
  "comment": "30 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We fix a positive integer $M$, and we consider expansions in arbitrary real bases $q>1$ over the alphabet $\\{0,1,...,M\\}$. We denote by $U_q$ the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension $D(q)$ of $U_q$ for each $q\\in (1,\\infty)$. Furthermore, we prove that the dimension function $D:(1,\\infty)\\to[0,1]$ is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in $(q',\\infty)$, where $q'$ denotes the Komornik--Loreti constant: although $D(q)>D(q')$ for all $q>q'$, we have $D'<0$ a.e. in $(q',\\infty)$. During the proofs we improve and generalize a theorem of Erd\\H{o}s et al. on the existence of large blocks of zeros in $\\beta$-expansions, and we determine for all $M$ the Lebesgue measure and the Hausdorff dimension of the set of bases in which $x=1$ has a unique expansion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-02T10:37:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63"
    ],
    "keywords": [
      "hausdorff dimension",
      "univoque sets",
      "unique expansion",
      "arbitrary real bases",
      "devils staircase behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}