{
  "id": "1512.03130",
  "version": "v1",
  "published": "2015-12-10T02:53:09.000Z",
  "updated": "2015-12-10T02:53:09.000Z",
  "title": "Every finite subset of an abelian group is an asymptotic approximate group",
  "authors": [
    "Melvyn B. Nathanson"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "If $A$ is a nonempty subset of an additive group $G$, then the $h$-fold sumset is \\[ hA = \\{x_1 + \\cdots + x_h : x_i \\in A_i \\text{ for } i=1,2,\\ldots, h\\}. \\] The set $A$ is an $(r,\\ell)$-approximate group in $G$ if $A$ is a nonempty subset of a group $G$ and there exists a subset $X$ of $G$ such that $|X| \\leq \\ell$ and $rA \\subseteq XA$. We do not assume that $A$ contains the identity, nor that $A$ is symmetric, nor that $A$ is finite. The set $A$ is an asymptotic $(r,\\ell)$-approximate group if the sumset $hA$ is an $(r,\\ell)$-approximate group for all sufficiently large $h$. It is proved that every polytope in a real vector space is an asymptotic $(r,\\ell)$-approximate group, that every finite set of lattice points is an asymptotic $(r,\\ell)$-approximate group, and that every finite subset of an abelian group is an asymptotic $(r,\\ell)$-approximate group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-10T02:53:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "05A18",
      "11B75",
      "11P70",
      "20K99",
      "20M99",
      "52B20"
    ],
    "keywords": [
      "asymptotic approximate group",
      "finite subset",
      "abelian group",
      "nonempty subset",
      "real vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151203130N"
    }
  }
}