{
  "id": "1510.05998",
  "version": "v1",
  "published": "2015-10-20T18:14:08.000Z",
  "updated": "2015-10-20T18:14:08.000Z",
  "title": "Extractors in Paley graphs: a random model",
  "authors": [
    "Rudi Mrazović"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "A well-known conjecture in analytic number theory states that for every pair of sets $X,Y\\subset\\mathbb{Z}/p\\mathbb{Z}$, each of size at least $\\log ^C p$ (for some constant $C$) we have that for $(\\frac12+o(1))|X||Y|$ of the pairs $(x,y)\\in X\\times Y$, $x+y$ is a quadratic residue modulo $p$. We address the probabilistic analogue of this question, that is for every fixed $\\delta>0$, given a finite group $G$ and $A\\subset G$ a random subset of density $\\frac12$, we prove that with high probability for all subsets $|X|,|Y|\\geq \\log ^{2+\\delta} |G|$ for $(\\frac12+o(1))|X||Y|$ of the pairs $(x,y)\\in X\\times Y$ we have $xy\\in A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-20T18:14:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random model",
      "paley graphs",
      "extractors",
      "analytic number theory states",
      "quadratic residue modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}