{
  "id": "2109.11003",
  "version": "v1",
  "published": "2021-09-22T19:46:14.000Z",
  "updated": "2021-09-22T19:46:14.000Z",
  "title": "Rational approximations of irrational numbers",
  "authors": [
    "Dimitris Koukoulopoulos"
  ],
  "comment": "20 pages; submitted to the Proceedings of the 2022 International Congress of Mathematicians",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Given quantities $\\Delta_1,\\Delta_2,\\dots\\geqslant 0$, a fundamental problem in Diophantine approximation is to understand which irrational numbers $x$ have infinitely many reduced rational approximations $a/q$ such that $|x-a/q|<\\Delta_q$. Depending on the choice of $\\Delta_q$ and of $x$, this question may be very hard. However, Duffin and Schaeffer conjectured in 1941 that if we assume a \"metric\" point of view, the question is governed by a simple zero--one law: writing $\\varphi$ for Euler's totient function, we either have $\\sum_{q=1}^\\infty \\varphi(q)\\Delta_q=\\infty$ and then almost all irrational numbers (in the Lebesgue sense) are approximable, or $\\sum_{q=1}^\\infty\\varphi(q)\\Delta_q<\\infty$ and almost no irrationals are approximable. We present the history of the Duffin--Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos--Maynard that settled it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-22T19:46:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irrational numbers",
      "simple zero-one law",
      "eulers totient function",
      "fundamental problem",
      "diophantine approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}