{
  "id": "1906.02029",
  "version": "v1",
  "published": "2019-06-05T13:45:17.000Z",
  "updated": "2019-06-05T13:45:17.000Z",
  "title": "Variants of Khintchine's theorem in metric Diophantine approximation",
  "authors": [
    "Laima Kaziulytė"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "New results towards the Duffin-Schaeffer conjecture, which is a fundamental unsolved problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function $\\psi: \\mathbb{N}\\to\\mathbb{R}$ we denote by $W(\\psi)$ the set of all $x\\in\\mathbb{R}$ such that $|nx-a|<\\psi(n)$ for infinitely many $a,n$. Analogously, denote $W'(\\psi)$ if we additionally require $a,n$ to be coprime. Aistleitner et al. ${\\tt arXiv:1803.05703 [math.NT]}$ proved that $W'(\\psi)$ is of full Lebesgue measure if there exist an $\\varepsilon>0$ such that $\\sum_{n=2}^\\infty\\psi(n)\\varphi(n)/(n(\\log n)^\\varepsilon)=\\infty$. This result seems to be the best one can expect from the method used. Assuming the extra divergence $\\sum_{n=2}^\\infty\\psi(n)/(\\log n)^\\varepsilon=\\infty$ we prove that $W(\\psi)$ is of full measure. This could also be deduced from the result in ${\\tt arXiv:1803.05703 [math.NT]}$, but we believe that our proof is of independent interest, since its method is totally different from the one in ${\\tt arXiv:1803.05703 [math.NT]}$. As a further application of our method, we prove that a variant of Khintchine's theorem is true without monotonicity, subject to an additional condition on the set of divisors of the support of $\\psi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-05T13:45:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "metric diophantine approximation",
      "khintchines theorem",
      "metric number theory",
      "full lebesgue measure",
      "additional condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}