{
  "id": "2110.09053",
  "version": "v2",
  "published": "2021-10-18T06:49:10.000Z",
  "updated": "2023-07-22T16:41:48.000Z",
  "title": "Difference sets in $\\mathbb{R}^d$",
  "authors": [
    "David Conlon",
    "Jeck Lim"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $d \\geq 2$ be a natural number. We show that $$|A-A| \\geq \\left(2d-2 + \\frac{1}{d-1}\\right)|A|-(2d^2-4d+3)$$ for any sufficiently large finite subset $A$ of $\\mathbb{R}^d$ that is not contained in a translate of a hyperplane. By a construction of Stanchescu, this is best possible and thus resolves an old question first raised by Uhrin.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-07-22T16:41:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B30",
      "11P70"
    ],
    "keywords": [
      "difference sets",
      "sufficiently large finite subset",
      "old question first",
      "natural number",
      "hyperplane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}