{
  "id": "0708.2853",
  "version": "v1",
  "published": "2007-08-21T14:38:12.000Z",
  "updated": "2007-08-21T14:38:12.000Z",
  "title": "Dense sets of integers with prescribed representation functions",
  "authors": [
    "Javier Cilleruelo",
    "Melvyn B. Nathanson"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let A be a set of integers and let h \\geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to the set A, and a_1\\leq ... \\leq a_h. The function r_{A,h} from the integers Z to the nonnegative integers N_0 U {\\infty} is called the representation function of order h for the set A. We prove that every function f from Z to N_0 U {\\infty} satisfying liminf_{|n|->\\infty} f (n)\\geq g is the representation function of order h for some sequence A of integers, and that A can be constructed so that it increases \"almost\" as slowly as any given B_h[g] sequence. In particular, for every epsilon >0 and g \\geq g(h,epsilon), we can construct a sequence A satisfying r_{A,h}=f and A(x)\\gg x^{(1/h)-epsilon}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-08-21T14:38:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B34",
      "11B05",
      "11A07",
      "11A41"
    ],
    "keywords": [
      "prescribed representation functions",
      "dense sets",
      "nonnegative integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.2853C"
    }
  }
}