{
  "id": "2106.06572",
  "version": "v1",
  "published": "2021-06-11T18:47:33.000Z",
  "updated": "2021-06-11T18:47:33.000Z",
  "title": "Hausdorff dimension of Gauss--Cantor sets and two applications to classical Lagrange and Markov spectra",
  "authors": [
    "Carlos Matheus",
    "Carlos Gustavo Moreira",
    "Mark Pollicott",
    "Polina Vytnova"
  ],
  "comment": "47 pages, 4 figures",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem \\ref{t.A}, provides a rigorous estimate on the smallest value $t_1$ such that the portion of the Markov spectrum $(-\\infty,t_1)\\cap M$ has Hausdorff dimension $1$. Our second result, Theorem \\ref{t.B}, gives a new upper bound on the Hausdorff dimension of the set difference $M\\setminus L$. Our method combines new facts about the structure of the classical spectra together with finer estimates on the Hausdorff dimension of Gauss--Cantor sets of continued fraction expansions whose entries satisfy appropriate restrictions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-11T18:47:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff dimension",
      "markov spectrum",
      "gauss-cantor sets",
      "classical lagrange",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}