{
  "id": "1307.8135",
  "version": "v1",
  "published": "2013-07-30T20:14:33.000Z",
  "updated": "2013-07-30T20:14:33.000Z",
  "title": "Irrational numbers associated to sequences without geometric progressions",
  "authors": [
    "Melvyn B. Nathanson",
    "Kevin O'Bryant"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let s and k be integers with s \\geq 2 and k \\geq 2. Let g_k^{(s)}(n) denote the cardinality of the largest subset of the set {1,2,..., n} that contains no geometric progression of length k whose common ratio is a power of s. Let r_k(\\ell) denote the cardinality of the largest subset of the set {0,1,2,\\ldots, \\ell -1\\} that contains no arithmetric progression of length k. The limit \\[ \\lim_{n\\rightarrow \\infty} \\frac{g_k^{(s)}(n)}{n} = (s-1) \\sum_{m=1}^{\\infty} \\left(\\frac{1}{s} \\right)^{\\min \\left(r_k^{-1}(m)\\right)} \\] exists and converges to an irrational number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-30T20:14:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B05",
      "11B25",
      "11B75",
      "11B83",
      "05D10",
      "11J99"
    ],
    "keywords": [
      "geometric progression",
      "irrational numbers",
      "largest subset",
      "cardinality",
      "common ratio"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.8135N"
    }
  }
}