{
  "id": "1112.3468",
  "version": "v1",
  "published": "2011-12-15T10:17:53.000Z",
  "updated": "2011-12-15T10:17:53.000Z",
  "title": "Contractions and expansion",
  "authors": [
    "Emmanuel Breuillard",
    "Ben Green"
  ],
  "comment": "6 pages, submitted to special volume of EJC in honour of Yahya Hamidoune",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let A be a finite set of reals and let K >= 1 be a real number. Suppose that for each a in A we are given an injective map f_a : A -> R which fixes a and contracts other points towards it in the sense that |a - f_a(x)| <= |a - x|/K for all x in A, and such that f_a(x) always lies between a and x. Then the union of the f_a(A) has cardinality >= K|A|/10 - O_K(1). An immediate consequence of this is the estimate |A + K.A| >= K|A|/10 - O_K(1), which is a slightly weakened version of a result of Bukh.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-15T10:17:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "contractions",
      "finite set",
      "real number",
      "immediate consequence",
      "injective map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.3468B"
    }
  }
}