{
  "id": "1907.04593",
  "version": "v1",
  "published": "2019-07-10T09:44:50.000Z",
  "updated": "2019-07-10T09:44:50.000Z",
  "title": "On the Duffin-Schaeffer conjecture",
  "authors": [
    "Dimitris Koukoulopoulos",
    "James Maynard"
  ],
  "comment": "45 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\psi:\\mathbb{N}\\to\\mathbb{R}_{\\ge0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\\mathcal{A}$ of real numbers $\\alpha$ for which there are infinitely many reduced fractions $a/q$ such that $|\\alpha-a/q|\\le \\psi(q)/q$. If $\\sum_{q=1}^\\infty \\psi(q)\\phi(q)/q=\\infty$, we show that $\\mathcal{A}$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $|\\alpha - a/q|\\le \\psi(q)/q$, giving a refinement of Khinchin's Theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-10T09:44:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83",
      "05C40"
    ],
    "keywords": [
      "duffin-schaeffer conjecture",
      "full lebesgue measure",
      "real numbers",
      "khinchins theorem",
      "arbitrary function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}