{
  "id": "2104.09239",
  "version": "v1",
  "published": "2021-04-19T12:26:18.000Z",
  "updated": "2021-04-19T12:26:18.000Z",
  "title": "Combinatorial structure of Sturmian words and continued fraction expansions of Sturmian numbers",
  "authors": [
    "Yann Bugeaud",
    "Michel Laurent"
  ],
  "comment": "44 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\theta = [0; a_1, a_2, \\dots]$ be the continued fraction expansion of an irrational real number $\\theta \\in (0, 1)$. It is well-known that the characteristic Sturmian word of slope $\\theta$ is the limit of a sequence of finite words $(M_k)_{k \\ge 0}$, with $M_k$ of length $q_k$ (the denominator of the $k$-th convergent to $\\theta$) being a suitable concatenation of $a_k$ copies of $M_{k-1}$ and one copy of $M_{k-2}$. Our first result extends this to any Sturmian word. Let $b \\ge 2$ be an integer. Our second result gives the continued fraction expansion of any real number $\\xi$ whose $b$-ary expansion is a Sturmian word ${\\bf s}$ over the alphabet $\\{0, b-1\\}$. This extends a classical result of B\\\"ohmer who considered only the case where ${\\bf s}$ is characteristic. As a consequence, we obtain a formula for the irrationality exponent of $\\xi$ in terms of the slope and the intercept of ${\\bf s}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-19T12:26:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J70",
      "11J82",
      "68R15"
    ],
    "keywords": [
      "continued fraction expansion",
      "sturmian numbers",
      "combinatorial structure",
      "irrational real number",
      "first result extends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}