{
  "id": "2502.09539",
  "version": "v1",
  "published": "2025-02-13T17:56:24.000Z",
  "updated": "2025-02-13T17:56:24.000Z",
  "title": "Erdős's integer dilation approximation problem and GCD graphs",
  "authors": [
    "Dimitris Koukoulopoulos",
    "Youness Lamzouri",
    "Jared Duker Lichtman"
  ],
  "comment": "47 pages, 1 figure",
  "categories": [
    "math.NT",
    "math.CO",
    "math.DS"
  ],
  "abstract": "Let $\\mathcal{A}\\subset\\mathbb{R}_{\\geqslant1}$ be a countable set such that $\\limsup_{x\\to\\infty}\\frac{1}{\\log x}\\sum_{\\alpha\\in\\mathcal{A}\\cap[1,x]}\\frac{1}{\\alpha}>0$. We prove that, for every $\\varepsilon>0$, there exist infinitely many pairs $(\\alpha, \\beta)\\in \\mathcal{A}^2$ such that $\\alpha\\neq \\beta$ and $|n\\alpha-\\beta| <\\varepsilon$ for some positive integer $n$. This resolves a problem of Erd\\H{o}s from 1948. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by the first author and by James Maynard in their work on the Duffin--Schaeffer conjecture in Diophantine approximation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-13T17:56:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "erdőss integer dilation approximation problem",
      "gcd graphs",
      "duffin-schaeffer conjecture",
      "diophantine approximation",
      "james maynard"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}