{
  "id": "2106.11628",
  "version": "v1",
  "published": "2021-06-22T09:35:20.000Z",
  "updated": "2021-06-22T09:35:20.000Z",
  "title": "The spectrum of the exponents of repetition",
  "authors": [
    "Deokwon Sim"
  ],
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "For an infinite word $\\mathbf{x}$, Bugeaud and Kim introduced a new complexity function $\\text{rep}(\\mathbf{x})$ which is called the exponent of repetition of $\\mathbf{x}$. They showed $1\\le \\text{rep}(\\mathbf{x}) \\le \\sqrt{10}-\\frac{3}{2}$ for any Sturmian word $\\mathbf{x}$. Ohnaka and Watanabe found a gap in the set of the exponents of repetition of Sturmian words. For an irrational number $\\theta\\in(0,1)$, let \\[ \\mathscr{L}(\\theta):=\\{\\text{rep}(\\mathbf{x}):\\textrm{$\\mathbf{x}$ is an Sturmian word of slope $\\theta$}\\}.\\] In this article, we look into $\\mathscr{L}(\\theta)$. The minimum of $\\mathscr{L}(\\theta)$ is determined where $\\theta$ has bounded partial quotients in its continued fraction expression. In particular, we found out the maximum and the minimum of $\\mathscr{L}(\\varphi)$ where $\\varphi:=\\frac{\\sqrt{5}-1}{2}$ is the fraction part of the golden ratio. Furthermore, we show that the three largest values are isolated points in $\\mathscr{L}(\\varphi)$ and the fourth largest point is a limit point of $\\mathscr{L}(\\varphi)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-22T09:35:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sturmian word",
      "repetition",
      "fourth largest point",
      "complexity function",
      "largest values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}