{
  "id": "1611.07269",
  "version": "v1",
  "published": "2016-11-22T12:22:29.000Z",
  "updated": "2016-11-22T12:22:29.000Z",
  "title": "More on the $h$-critical numbers of finite abelian groups",
  "authors": [
    "Bela Bajnok"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a finite abelian group $G$, a nonempty subset $A$ of $G$, and a positive integer $h$, we let $hA$ denote the $h$-fold sumset of $A$; that is, $hA$ is the collection of sums of $h$ not-necessarily-distinct elements of $A$. Furthermore, for a positive integer $s$, we set $[0,s] A=\\cup_{h=0}^s h A$. We say that $A$ is a generating set of $G$ if there is a positive integer $s$ for which $[0,s] A=G$. The $h$-critical number $\\chi (G,h)$ of $G$ is defined as the smallest positive integer $m$ for which $hA=G$ holds for every $m$-subset $A$ of $G$; similarly, $\\chi (G,[0,s])$ is the smallest positive integer $m$ for which $[0,s]A=G$ holds for every $m$-subset $A$ of $G$. We define $\\widehat{\\chi} (G, h)$ as the smallest positive integer $m$ for which $hA=G$ holds for every generating $m$-subset $A$ of $G$; $\\widehat{\\chi} (G, [0,s])$ is defined similarly. The value of $\\chi (G,h)$ has been determined by this author for all $G$ and $h$, and $\\widehat{\\chi} (G, [0,s])$ was introduced and resolved for some special cases by Klopsch and Lev. Here we determine the remaining two quantities in all cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-22T12:22:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite abelian group",
      "critical number",
      "smallest positive integer",
      "not-necessarily-distinct elements",
      "special cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}