{
  "id": "0902.0488",
  "version": "v4",
  "published": "2009-02-03T12:25:46.000Z",
  "updated": "2009-12-23T15:19:32.000Z",
  "title": "Growth rate for beta-expansions",
  "authors": [
    "De-Jun Feng",
    "Nikita Sidorov"
  ],
  "comment": "21 pages, 2 figures",
  "journal": "Monatsh. Math. 162 (2011), 41-60",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $\\beta>1$ and let $m>\\be$ be an integer. Each $x\\in I_\\be:=[0,\\frac{m-1}{\\beta-1}]$ can be represented in the form \\[ x=\\sum_{k=1}^\\infty \\epsilon_k\\beta^{-k}, \\] where $\\epsilon_k\\in\\{0,1,...,m-1\\}$ for all $k$ (a $\\beta$-expansion of $x$). It is known that a.e. $x\\in I_\\beta$ has a continuum of distinct $\\beta$-expansions. In this paper we prove that if $\\beta$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $\\beta$. When $\\beta<\\frac{1+\\sqrt5}2$, we show that the set of $\\beta$-expansions grows exponentially for every internal $x$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-12-23T15:19:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "28D05",
      "42A85"
    ],
    "keywords": [
      "growth rate",
      "beta-expansions",
      "lebesgue-generic local dimension",
      "pisot number",
      "expansions grows"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.0488F"
    }
  }
}