{
  "id": "1210.4203",
  "version": "v5",
  "published": "2012-10-15T21:42:36.000Z",
  "updated": "2013-09-19T20:48:51.000Z",
  "title": "A Cauchy-Davenport theorem for semigroups",
  "authors": [
    "Salvatore Tringali"
  ],
  "comment": "14 pages, to appear in Uniform Distribution Theory. Fixed minor details w.r.t. the previous version",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "We generalize the Davenport transform and use it to prove that, for a (possibly non-commutative) cancellative semigroup $\\mathbb A = (A, +)$ and non-empty subsets $X,Y$ of $A$ such that the subsemigroup generated by $Y$ is commutative, we have $|X + Y| \\ge \\min(\\omega(Y), |X| + |Y| - 1)$, where $\\omega(Y) := \\sup_{y_0 \\in Y \\cap \\mathbb A^{\\times}} \\inf_{y \\in Y \\setminus \\{y_0\\}} |<y - y_0>|$. This carries over the Cauchy-Davenport theorem to the broader setting of semigroups, and it implies, in particular, an extension of I. Chowla's and S.S. Pillai's theorems for cyclic groups and a notable strengthening of another generalization of the same Cauchy-Davenport theorem to commutative groups, where $\\omega(Y)$ in the above is replaced by the minimal order of the non-trivial subgroups of $\\mathbb A$.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2013-09-19T20:48:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15",
      "11B13",
      "20E99",
      "20M10"
    ],
    "keywords": [
      "cauchy-davenport theorem",
      "non-empty subsets",
      "davenport transform",
      "pillais theorems",
      "cyclic groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.4203T"
    }
  }
}