{
  "id": "2401.05843",
  "version": "v1",
  "published": "2024-01-11T11:26:09.000Z",
  "updated": "2024-01-11T11:26:09.000Z",
  "title": "A Non-Trivial Minoration for the Set of Salem Numbers",
  "authors": [
    "Jean-Louis Verger-Gaugry"
  ],
  "comment": "text dedicated to the 75th birthday of Christiane Frougny - NUMERATION2023 -. arXiv admin note: text overlap with arXiv:1911.10590",
  "categories": [
    "math.NT"
  ],
  "abstract": "The set of Salem numbers is proved to be bounded from below by $\\theta_{31}^{-1}= 1.08544\\ldots$ where $\\theta_{n}$, $ n \\geq 2$, is the unique root in $(0,1)$ of the trinomial $-1+x+x^n$. Lehmer's number $1.176280\\ldots$ belongs to the interval $(\\theta_{12}^{-1}, \\theta_{11}^{-1})$. We conjecture that there is no Salem number in $(\\theta_{31}^{-1}, \\theta_{12}^{-1}) = (1.08544\\ldots, 1.17295\\ldots)$. For proving the Main Theorem, the algebraic and analytic properties of the dynamical zeta function of the R\\'enyi-Parry numeration system are used, with real bases running over the set of real reciprocal algebraic integers, and variable tending to 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-11T11:26:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K16",
      "11M41",
      "11R06",
      "11R09",
      "30B10",
      "30B40",
      "37C30",
      "37N99",
      "03D45"
    ],
    "keywords": [
      "salem number",
      "non-trivial minoration",
      "real reciprocal algebraic integers",
      "renyi-parry numeration system",
      "lehmers number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}