{
  "id": "math/0209247",
  "version": "v1",
  "published": "2002-09-19T14:38:23.000Z",
  "updated": "2002-09-19T14:38:23.000Z",
  "title": "Universal $β$-expansions",
  "authors": [
    "Nikita Sidorov"
  ],
  "comment": "11 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Given $\\beta\\in(1,2)$, a $\\beta$-expansion of a real $x$ is a power series in base $\\beta$ with coefficients 0 and 1 whose sum equals $x$. The aim of this note is to study certain problems related to the universality and combinatorics of $\\beta$-expansions. Our main result is that for any $\\beta\\in(1,2)$ and a.e. $x\\in (0,1)$ there always exists a universal $\\beta$-expansion of $x$ in the sense of Erd\\\"os and Komornik, i.e., a $\\beta$-expansion whose complexity function is $2^n$. We also study some questions related to the points having less than a full branching continuum of $\\beta$-expansions and also normal $\\beta$-expansions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-09-19T14:38:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "11K16",
      "28D05"
    ],
    "keywords": [
      "power series",
      "sum equals",
      "main result",
      "complexity function",
      "full branching continuum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......9247S"
    }
  }
}