{
  "id": "1412.4058",
  "version": "v1",
  "published": "2014-12-12T17:12:06.000Z",
  "updated": "2014-12-12T17:12:06.000Z",
  "title": "The $h$-critical number of finite abelian groups",
  "authors": [
    "Bela Bajnok"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $\\chi(G,h)$ (resp.~$\\chi \\hat{\\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold unrestricted (resp.~restricted) sumset of every $m$-subset of $G$ equals $G$ itself. Here we determine $\\chi(G,h)$ for all $G$ and $h$; and prove several results for $\\chi \\hat{\\;}(G,h)$, including the cases of any $G$ and $h = 2$, any $G$ and large $h$, and any $h$ for the cyclic group $\\mathbb{Z}_n$ of even order. We also provide a lower bound for $\\chi \\hat{\\;}(\\mathbb{Z}_n,3)$ that we believe is exact for every $n$---this conjecture is a generalization of the one made by Gallardo, Grekos, et al.~that was proved (for large $n$) by Lev.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-12T17:12:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite abelian group",
      "critical number",
      "minimum value",
      "cyclic group",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.4058B"
    }
  }
}