{
  "id": "2005.04663",
  "version": "v1",
  "published": "2020-05-10T13:29:25.000Z",
  "updated": "2020-05-10T13:29:25.000Z",
  "title": "On maximal product sets of random sets",
  "authors": [
    "Daniele Mastrostefano"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "For every positive integer N and every $\\alpha\\in [0,1)$, let $B(N, \\alpha)$ denote the probabilistic model in which a random set $A\\subset \\{1,\\dots,N\\}$ is constructed by choosing independently every element of $\\{1,\\dots,N\\}$ with probability $\\alpha$. We prove that, as $N\\longrightarrow +\\infty$, for every $A$ in $B(N, \\alpha)$ we have $|AA|\\ \\sim |A|^2/2$ with probability $1-o(1)$, if and only if $$\\frac{\\log(\\alpha^2(\\log N)^{\\log 4-1})}{\\sqrt{\\log\\log N}}\\longrightarrow-\\infty.$$ This improves a theorem of Cilleruelo, Ramana and Ramar\\'e, who proved the above asymptotic between $|AA|$ and $|A|^2/2$ when $\\alpha=o(1/\\sqrt{\\log N})$, and supplies a complete characterization of maximal product sets of random sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-10T13:29:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B99"
    ],
    "keywords": [
      "maximal product sets",
      "random set",
      "probabilistic model",
      "probability",
      "complete characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}