{
  "id": "2106.04036",
  "version": "v1",
  "published": "2021-06-08T01:00:46.000Z",
  "updated": "2021-06-08T01:00:46.000Z",
  "title": "Ratio sets of random sets",
  "authors": [
    "Javier Cilleruelo",
    "Jorge Guijarro-Ordonez"
  ],
  "journal": "The Ramanujan Journal, 43(2), 2017",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We study the typical behavior of the size of the ratio set $A/A$ for a random subset $A\\subset \\{1,\\dots , n\\}$. For example, we prove that $|A/A|\\sim \\frac{2\\text{Li}_2(3/4)}{\\pi^2}n^2 $ for almost all subsets $A \\subset\\{1,\\dots ,n\\}$. We also prove that the proportion of visible lattice points in the lattice $A_1\\times\\cdots \\times A_d$, where $A_i$ is taken at random in $[1,n]$ with $\\mathbb P(m\\in A_i)=\\alpha_i$ for any $m\\in [1,n]$, is asymptotic to a constant $\\mu(\\alpha_1,\\dots,\\alpha_d)$ that involves the polylogarithm of order $d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-08T01:00:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ratio set",
      "random sets",
      "random subset",
      "visible lattice points",
      "typical behavior"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}