{
  "id": "1009.6140",
  "version": "v1",
  "published": "2010-09-30T14:00:52.000Z",
  "updated": "2010-09-30T14:00:52.000Z",
  "title": "On the cardinality of sumsets in torsion-free groups",
  "authors": [
    "Károly J. Böröczky",
    "Péter P. Pálfy",
    "Oriol Serra"
  ],
  "doi": "10.1112/blms/bds032",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Let $A, B$ be finite subsets of a torsion-free group $G$. We prove that for every positive integer $k$ there is a $c(k)$ such that if $|B|\\ge c(k)$ then the inequality $|AB|\\ge |A|+|B|+k$ holds unless a left translate of $A$ is contained in a cyclic subgroup. We obtain $c(k)<c_0k^{6}$ for arbitrary torsion-free groups, and $c(k)<c_0k^{3}$ for groups with the unique product property, where $c_0$ is an absolute constant. We give examples to show that $c(k)$ is at least quadratic in $k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-30T14:00:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cardinality",
      "unique product property",
      "arbitrary torsion-free groups",
      "absolute constant",
      "finite subsets"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.6140B"
    }
  }
}