{
  "id": "1612.01929",
  "version": "v1",
  "published": "2016-12-06T18:03:31.000Z",
  "updated": "2016-12-06T18:03:31.000Z",
  "title": "Sumsets as unions of sumsets of subsets",
  "authors": [
    "Jordan S. Ellenberg"
  ],
  "comment": "3 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $S$ and $T$ be subsets of $\\mathbf{F}_q^n$. We show there are subsets $S'$ of $S$ and $T'$ of $T$ such that $S+T$ is the union of $S+T'$ and $S'+T$, with $|S'| + |T'|$ bounded by $c^n$ with $c < q$. The proof relies on the method of Croot-Lev-Pach and Ellenberg-Gijswijt on the cap set problem, together with a result of Meshulam on linear spaces of low-rank matrices. The result is a modest generalization of the recent bounds on (single-colored and multi-colored) sum-free sets by the author and others.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-06T18:03:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cap set problem",
      "proof relies",
      "modest generalization",
      "low-rank matrices",
      "linear spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}