{
  "id": "1511.06478",
  "version": "v1",
  "published": "2015-11-20T03:02:10.000Z",
  "updated": "2015-11-20T03:02:10.000Z",
  "title": "Every finite set of integers is an asymptotic approximate group",
  "authors": [
    "Melvyn B. Nathanson"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "A set $A$ is an $(r,\\ell)$-approximate group in the additive abelian group $G$ if $A$ is a nonempty subset of $G$ and there exists a subset $X$ of $G$ such that $|X| \\leq \\ell$ and $rA \\subseteq X+A$. The set $A$ is an asymptotic $(r,\\ell)$-approximate group if the sumset $hA$ is an $(r,\\ell)$-approximate group for all sufficiently large integers $h$. It is proved that every finite set of integers is an asymptotic $(r,r+1)$-approximate group for every integer $r \\geq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-20T03:02:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "05A17",
      "11B75",
      "11P99"
    ],
    "keywords": [
      "asymptotic approximate group",
      "finite set",
      "additive abelian group",
      "nonempty subset",
      "sufficiently large integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151106478N"
    }
  }
}