{
  "id": "0903.2714",
  "version": "v1",
  "published": "2009-03-16T09:59:48.000Z",
  "updated": "2009-03-16T09:59:48.000Z",
  "title": "The number of rational numbers determined by large sets of integers",
  "authors": [
    "Javier Cilleruelo",
    "D. S. Ramana",
    "Olivier Ramare"
  ],
  "comment": "11 pages",
  "doi": "10.1112/blms/bdq021",
  "categories": [
    "math.NT"
  ],
  "abstract": "When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \\geq \\alpha X$ and $|B| \\geq \\beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \\times B$ is $\\gg (\\alpha \\beta)^{1+\\epsilon}XY$ for any $\\epsilon > 0$, where the implied constant depends on $\\epsilon$ alone. We then construct examples that show that this bound cannot in general be improved to $\\gg \\alpha \\beta XY$. We also resolve the natural generalisation of our problem to arbitrary subsets $C$ of the integer points in $[1,X] \\times [1,Y]$. Finally, we apply our results to answer a question of S\\'ark\\\"ozy concerning the differences of consecutive terms of the product sequence of a given integer sequence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-16T09:59:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B05"
    ],
    "keywords": [
      "rational numbers",
      "large sets",
      "product sequence",
      "implied constant depends",
      "integer points"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.2714C"
    }
  }
}