{
  "id": "2005.04371",
  "version": "v1",
  "published": "2020-05-09T05:39:39.000Z",
  "updated": "2020-05-09T05:39:39.000Z",
  "title": "Diophantine approximation by negative continued fraction",
  "authors": [
    "Hiroaki Ito"
  ],
  "comment": "8 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ \\frac{\\log{n}}{n}\\log{\\left|x-\\frac{P_n}{Q_n}\\right|}\\rightarrow -\\frac{\\pi^2}{3} \\quad \\text{in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-09T05:39:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diophantine approximation",
      "limit superior",
      "growth rate",
      "arithmetic mean",
      "negative continued fraction converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}