{
  "id": "1507.07672",
  "version": "v1",
  "published": "2015-07-28T07:52:07.000Z",
  "updated": "2015-07-28T07:52:07.000Z",
  "title": "If $(A+A)/(A+A)$ is small then the ratio set is large",
  "authors": [
    "Oliver Roche-Newton"
  ],
  "comment": "21 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In this paper, we consider the sum-product problem of obtaining lower bounds for the size of the set $$\\frac{A+A}{A+A}:=\\left \\{ \\frac{a+b}{c+d} : a,b,c,d \\in A, c+d \\neq 0 \\right\\},$$ for an arbitrary finite set $A$ of real numbers. The main result is the bound $$\\left| \\frac{A+A}{A+A} \\right| \\gg \\frac{|A|^{2+\\frac{1}{15}}}{|A:A|^{\\frac{1}{30}}\\log |A|},$$ where $A:A$ denotes the ratio set of $A$. This improves on a result of Balog and the author (arXiv:1402.5775), provided that the size of the ratio set is subquadratic in $|A|$. That is, we establish that the inequality $$\\left| \\frac{A+A}{A+A} \\right| \\ll |A|^{2} \\Rightarrow |A:A| \\gg \\frac{ |A|^2}{\\log^{30}|A|} . $$ This extremal result answers a question similar to some conjectures in a recent paper of the author and Zhelezov (arXiv:1410.1156).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-28T07:52:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "11B75",
      "52C10"
    ],
    "keywords": [
      "ratio set",
      "arbitrary finite set",
      "extremal result answers",
      "question similar",
      "real numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150707672R"
    }
  }
}