{
  "id": "0712.0102",
  "version": "v2",
  "published": "2007-12-01T19:14:22.000Z",
  "updated": "2008-10-05T18:19:19.000Z",
  "title": "Univoque numbers and an avatar of Thue-Morse",
  "authors": [
    "Jean-Paul Allouche",
    "Christiane Frougny"
  ],
  "comment": "accepted by Acta Arithmetica",
  "journal": "Acta Arithmetica, 136 (2009) 319-329",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Univoque numbers are real numbers $\\lambda > 1$ such that the number 1 admits a unique expansion in base $\\lambda$, i.e., a unique expansion $1 = \\sum_{j \\geq 0} a_j \\lambda^{-(j+1)}$, with $a_j \\in \\{0, 1, ..., \\lceil \\lambda \\rceil -1\\}$ for every $j \\geq 0$. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called {\\em admissible sequences}. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set $\\{0, 1, >..., b\\}$, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval $(b, b+1)$, where $b$ is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism $3 \\to 31$, $2 \\to 30$, $1 \\to 03$, $0 \\to 02$, occurs in a \"universal\" manner.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-10-05T18:19:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "11B83",
      "11B85",
      "68R15",
      "11J81"
    ],
    "keywords": [
      "unique expansion",
      "smallest univoque number belonging",
      "smallest admissible sequence",
      "thue-morse sequence",
      "real numbers"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.4064/aa136-4-2",
      "journal": "Acta Arithmetica",
      "year": 2009,
      "volume": 136,
      "number": 4,
      "pages": 319
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009AcAri.136..319A"
    }
  }
}