{
  "id": "1412.1608",
  "version": "v1",
  "published": "2014-12-04T10:21:12.000Z",
  "updated": "2014-12-04T10:21:12.000Z",
  "title": "The Minimum Size of Signed Sumsets",
  "authors": [
    "Bela Bajnok",
    "Ryan Matzke"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\\rho(G, m, h) = \\min \\{|hA| \\; : \\; A \\subseteq G, |A|=m\\}$$ and $$\\rho_{\\pm} (G, m, h) = \\min \\{|h_{\\pm} A| \\; : \\; A \\subseteq G, |A|=m\\},$$ where $hA$ and $h_{\\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\\rho_{\\pm}(G, m, h)$ equals $\\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\\rho_{\\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m$, and $h$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-04T10:21:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimum size",
      "finite abelian group",
      "upper bound",
      "fold sumset",
      "fold signed sumset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.1608B"
    }
  }
}