{
  "id": "2207.07263",
  "version": "v1",
  "published": "2022-07-15T02:58:59.000Z",
  "updated": "2022-07-15T02:58:59.000Z",
  "title": "Univoque bases of real numbers: simply normal bases, irregular bases and multiple rationals",
  "authors": [
    "Yu Hu",
    "Yan Huang",
    "Derong Kong"
  ],
  "comment": "25 pages, 2 figures",
  "categories": [
    "math.DS",
    "math.CA",
    "math.NT"
  ],
  "abstract": "Given a positive integer $M$ and a real number $x\\in(0,1]$, we call $q\\in(1,M+1]$ a univoque simply normal base of $x$ if there exists a unique simply normal sequence $(d_i)\\in\\{0,1,\\ldots,M\\}^\\mathbb N$ such that $x=\\sum_{i=1}^\\infty d_i q^{-i}$. Similarly, a base $q\\in(1,M+1]$ is called a univoque irregular base of $x$ if there exists a unique sequence $(d_i)\\in\\{0,1,\\ldots, M\\}^\\mathbb N$ such that $x=\\sum_{i=1}^\\infty d_i q^{-i}$ and the sequence $(d_i)$ has no digit frequency. Let $\\mathcal U_{SN}(x)$ and $\\mathcal U_{I_r}(x)$ be the sets of univoque simply normal bases and univoque irregular bases of $x$, respectively. In this paper we show that for any $x\\in(0,1]$ both $\\mathcal U_{SN}(x)$ and $\\mathcal U_{I_r}(x)$ have full Hausdorff dimension. Furthermore, given finitely many rationals $x_1, x_2, \\ldots, x_n\\in(0,1]$ so that each $x_i$ has a finite expansion in base $M+1$, we show that there exists a full Hausdorff dimensional set of $q\\in(1,M+1]$ such that each $x_i$ has a unique expansion in base $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-15T02:58:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real number",
      "univoque bases",
      "multiple rationals",
      "univoque simply normal base",
      "univoque irregular base"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}