{
  "id": "1608.03209",
  "version": "v1",
  "published": "2016-08-10T15:22:39.000Z",
  "updated": "2016-08-10T15:22:39.000Z",
  "title": "When almost all sets are difference dominated in $\\mathbb{Z}/n\\mathbb{Z}$",
  "authors": [
    "Anand Hemmady",
    "Adam Lott",
    "Steven J. Miller"
  ],
  "comment": "Version 1.0, 13 pages",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "We investigate the behavior of the sum and difference sets of $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$ chosen independently and randomly according to a binomial parameter $p(n) = o(1)$. We show that for rapidly decaying $p(n)$, $A$ is almost surely difference-dominated as $n \\to \\infty$, but for slowly decaying $p(n)$, $A$ is almost surely balanced as $n \\to \\infty$, with a continuous phase transition as $p(n)$ crosses a critical threshold. Specifically, we show that if $p(n) = o(n^{-1/2})$, then $|A-A|/|A+A|$ converges to $2$ almost surely as $n \\to \\infty$ and if $p(n) = c \\cdot n^{-1/2}$, then $|A-A|/|A+A|$ converges to $1+\\exp(-c^2/2)$ almost surely as $n \\to \\infty$. In these cases, we modify the arguments of Hegarty and Miller on subsets of $\\mathbb{Z}$ to prove our results. When $\\sqrt{\\log n} \\cdot n^{-1/2} = o(p(n))$, we prove that $|A-A| = |A+A| = n$ almost surely as $n \\to \\infty$ if some additional restrictions are placed on $n$. In this case, the behavior is drastically different from that of subsets of $\\mathbb{Z}$ and new technical issues arise, so a novel approach is needed. When $n^{-1/2} = o(p(n))$ and $p(n) = o(\\sqrt{ \\log n} \\cdot n^{-1/2})$, the behavior of $|A+A|$ and $|A-A|$ is markedly different and suggests an avenue for further study. These results establish a \"correspondence principle\" with the existing results of Hegarty, Miller, and Vissuet. As $p(n)$ decays more rapidly, the behavior of subsets of $\\mathbb{Z}/n\\mathbb{Z}$ approaches the behavior of subsets of $\\mathbb{Z}$ shown by Hegarty and Miller. Moreover, as $p(n)$ decays more slowly, the behavior of subsets of $\\mathbb{Z}/n\\mathbb{Z}$ approaches the behavior shown by Miller and Vissuet in the case where $p(n) = 1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-10T15:22:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11P99",
      "05B10",
      "11K99"
    ],
    "keywords": [
      "difference sets",
      "continuous phase transition",
      "novel approach",
      "technical issues arise",
      "additional restrictions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}