{
  "id": "2403.16589",
  "version": "v1",
  "published": "2024-03-25T09:59:44.000Z",
  "updated": "2024-03-25T09:59:44.000Z",
  "title": "Sumsets in the Hypercube",
  "authors": [
    "Noga Alon",
    "Or Zamir"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A subset $S$ of the Boolean hypercube $\\mathbb{F}_2^n$ is a sumset if $S = A+A = \\{a + b \\ | \\ a, b\\in A\\}$ for some $A \\subseteq \\mathbb{F}_2^n$. We prove that the number of sumsets in $\\mathbb{F}_2^n$ is asymptotically $(2^n-1)2^{2^{n-1}}$. Furthermore, we show that the family of sumsets in $\\mathbb{F}_2^n$ is almost identical to the family of all subsets of $\\mathbb{F}_2^n$ that contain a complete linear subspace of co-dimension $1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-25T09:59:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complete linear subspace",
      "boolean hypercube",
      "co-dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}