{
  "id": "1704.04691",
  "version": "v1",
  "published": "2017-04-15T21:37:33.000Z",
  "updated": "2017-04-15T21:37:33.000Z",
  "title": "Fourier series and Diophantine approximation",
  "authors": [
    "Han Yu"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The Duffin-Schaeffer conjecture says that if $\\sum_{p}f(p)\\phi(p)/p=\\infty$, then for Lebesgue a.e number $x\\in [0,1]$, the inequality $|x-q/p|<f(p)/p$ holds for infinitely many coprime pairs $(p,q)$. In this paper, we introduce a Fourier analytic method to deal with this conjecture with extra logarithmic divergence, namely: \\[ \\frac{f(p)}{\\log ^C p}=\\infty. \\] We obtain a partial result from this extra divergence property which says that \\ds holds if we have the above divergence and the inequality $f(p)p>\\log ^B p$ either never holds or holds for sufficiently many $p$ in a precise sense. Here the dependence of numbers $C,B$ is that: $ C>(B+5)/2. $",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-15T21:37:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83",
      "11K55",
      "11K60"
    ],
    "keywords": [
      "diophantine approximation",
      "fourier series",
      "duffin-schaeffer conjecture says",
      "extra logarithmic divergence",
      "fourier analytic method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}