{
  "id": "1305.1685",
  "version": "v3",
  "published": "2013-05-08T00:51:39.000Z",
  "updated": "2013-09-04T00:37:34.000Z",
  "title": "A note on the Duffin-Schaeffer conjecture with slow divergence",
  "authors": [
    "Christoph Aistleitner"
  ],
  "comment": "4 pages; for version 2 some typos have been fixed and a corollary has been added; for version 3, some further minor changes have been made. The manuscript has been accepted for publication by Bull. London Math. Soc",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a non-negative function $\\psi: ~ \\N \\mapsto \\R$, let $W(\\psi)$ denote the set of real numbers $x$ for which the inequality $|n x - a| < \\psi(n)$ has infinitely many coprime solutions $(a,n)$. The Duffin--Schaeffer conjecture, one of the most important unsolved problems in metric number theory, asserts that $W(\\psi)$ has full measure provided {equation} \\label{dsccond} \\sum_{n=1}^\\infty \\frac{\\psi(n) \\varphi(n)}{n} = \\infty. {equation} Recently Beresnevich, Harman, Haynes and Velani proved that $W(\\psi)$ has full measure under the \\emph{extra divergence} condition $$ \\sum_{n=1}^\\infty \\frac{\\psi(n) \\varphi(n)}{n \\exp(c (\\log \\log n) (\\log \\log \\log n))} = \\infty \\qquad \\textrm{for some $c>0$}. $$ In the present note we establish a \\emph{slow divergence} counterpart of their result: $W(\\psi)$ has full measure, provided\\eqref{dsccond} holds and additionally there exists some $c>0$ such that $$ \\sum_{n=2^{2^h}+1}^{2^{2^{h+1}}} \\frac{\\psi(n) \\varphi(n)}{n} \\leq \\frac{c}{h} \\qquad \\textrm{for all \\quad $h \\geq 1$.} $$",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-04T00:37:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K60",
      "11J83"
    ],
    "keywords": [
      "duffin-schaeffer conjecture",
      "slow divergence",
      "full measure",
      "metric number theory",
      "important unsolved problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.1685A"
    }
  }
}