{
  "id": "math/0604055",
  "version": "v1",
  "published": "2006-04-04T02:24:13.000Z",
  "updated": "2006-04-04T02:24:13.000Z",
  "title": "Density of sets of natural numbers and the Levy group",
  "authors": [
    "Melvyn B. Nathanson",
    "Rohit Parikh"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\N$ denote the set of positive integers. The asymptotic density of the set $A \\subseteq \\N$ is $d(A) = \\lim_{n\\to\\infty} |A\\cap [1,n]|/n$, if this limit exists. Let $ \\mathcal{AD}$ denote the set of all sets of positive integers that have asymptotic density, and let $S_{\\N}$ denote the set of all permutations of the positive integers \\N. The group $\\mathcal{L}^{\\sharp}$ consists of all permutations $f \\in S_{\\N}$ such that $A \\in \\mathcal{AD}$ if and only if $f(A) \\in \\mathcal{AD}$, and the group $\\mathcal{L}^{\\ast}$ consists of all permutations $f \\in \\mathcal{L}^{\\sharp}$ such that $d(f(A)) = d(A)$ for all $A \\in \\mathcal{AD}$. Let $f:\\N \\to \\N $ be a one-to-one function such that $d(f(\\N))=1$ and, if $A \\in \\mathcal{AD}$, then $f(A) \\in \\mathcal{AD}$. It is proved that $f$ must also preserve density, that is, $d(f(A)) = d(A)$ for all $A \\in \\mathcal{AD}$. Thus, the groups $\\mathcal{L}^{\\sharp}$ and $\\mathcal{L}^{\\ast}$ coincide.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-04-04T02:24:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B05",
      "11B13",
      "11B75"
    ],
    "keywords": [
      "natural numbers",
      "levy group",
      "positive integers",
      "asymptotic density",
      "permutations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......4055N"
    }
  }
}