{
  "id": "1909.05188",
  "version": "v1",
  "published": "2019-09-11T16:37:37.000Z",
  "updated": "2019-09-11T16:37:37.000Z",
  "title": "A note on product sets of random sets",
  "authors": [
    "Carlo Sanna"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given two sets of positive integers $A$ and $B$, let $AB := \\{ab : a \\in A,\\, b \\in B\\}$ be their product set and put $A^k := A \\cdots A$ ($k$ times $A$) for any positive integer $k$. Moreover, for every positive integer $n$ and every $\\alpha \\in [0,1]$, let $\\mathcal{B}(n, \\alpha)$ denote the probabilistic model in which a random set $A \\subseteq \\{1, \\dots, n\\}$ is constructed by choosing independently every element of $\\{1, \\dots, n\\}$ with probability $\\alpha$. We prove that if $A_1, \\dots, A_s$ are random sets in $\\mathcal{B}(n_1, \\alpha_1), \\dots, \\mathcal{B}(n_s, \\alpha_s)$, respectively, $k_1, \\dots, k_s$ are fixed positive integers, $\\alpha_i n_i \\to +\\infty$, and $1/\\alpha_i$ does not grow too fast in terms of a product of $\\log n_j$; then $|A_1^{k_1} \\cdots A_s^{k_s}| \\sim \\frac{|A_1|^{k_1}}{k_1!}\\cdots\\frac{|A_s|^{k_s}}{k_s!}$ with probability $1 - o(1)$. This is a generalization of a result of Cilleruelo, Ramana, and Ramar\\'e, who considered the case $s = 1$ and $k_1 = 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-11T16:37:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N99"
    ],
    "keywords": [
      "random set",
      "product set",
      "probabilistic model",
      "probability",
      "fixed positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}