{
  "id": "2006.13356",
  "version": "v1",
  "published": "2020-06-23T22:04:13.000Z",
  "updated": "2020-06-23T22:04:13.000Z",
  "title": "A note on the natural density of product sets",
  "authors": [
    "Sandro Bettin",
    "Dimitris Koukoulopoulos",
    "Carlo Sanna"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Given two sets of natural numbers $\\mathcal{A}$ and $\\mathcal{B}$ of natural density $1$ we prove that their product set $\\mathcal{A}\\cdot \\mathcal{B}:=\\{ab:a\\in\\mathcal{A},\\,b\\in\\mathcal{B}\\}$ also has natural density $1$. On the other hand, for any $\\varepsilon>0$, we show there are sets $\\mathcal{A}$ of density $>1-\\varepsilon$ for which the product set $\\mathcal{A}\\cdot\\mathcal{A}$ has density $<\\varepsilon$. This answers two questions of Hegyv\\'{a}ri, Hennecart and Pach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-23T22:04:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "product set",
      "natural density",
      "natural numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}