{
  "id": "2102.13264",
  "version": "v1",
  "published": "2021-02-26T02:12:38.000Z",
  "updated": "2021-02-26T02:12:38.000Z",
  "title": "Large intersection of univoque bases of real numbers",
  "authors": [
    "Kan Jiang",
    "Derong Kong",
    "Wenxia Li"
  ],
  "comment": "28 pages, 5 figures",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Let $x\\in(0,1)$ and $m\\in\\mathbb N_{\\ge 2}$. We consider the set $\\Lambda(x)$ of bases $\\lambda\\in(0, 1/m]$ such that $x=\\sum_{i=1}^\\infty d_i \\lambda^i$ for some (unique) sequence $(d_i)\\in\\{0,1,\\ldots,m-1\\}^\\mathbb N$. In this paper we show that $\\Lambda(x)$ is a topological Cantor set; it has zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that the intersection $\\Lambda(x)\\cap\\Lambda(y)$ has full Hausdorff dimension for any $x, y\\in(0,1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-26T02:12:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real numbers",
      "univoque bases",
      "large intersection",
      "full hausdorff dimension",
      "zero lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}