{
  "id": "math/0302091",
  "version": "v2",
  "published": "2003-02-10T19:05:23.000Z",
  "updated": "2003-12-03T19:11:11.000Z",
  "title": "Every function is the representation function of an additive basis for the integers",
  "authors": [
    "Melvyn B. Nathanson"
  ],
  "comment": "15 pages. LaTex file. Corrected and revised manuscript",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let A be a set of integers. For every integer n, let r_{A,h}(n) denote the number of representations of n in the form n = a_1 + a_2 + ... + a_h, where a_1, a_2,...,a_h are in A and a_1 \\leq a_2 \\leq ... \\leq a_h. The function r_{A,h}: Z \\to N_0 \\cup \\infty is the representation function of order h for A. The set A is called an asymptotic basis of order h if r_{A,h}^{-1}(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of exactly h not necessarily distinct elements of A. It is proved that every function is a representation function, that is, if f: Z \\to N_0 \\cup \\infty is any function such that f^{-1}(0) is finite, then there exists a set A of integers such that f(n) = r_{A,h}(n) for all n in Z. Moreover, the set A can be arbitrarily sparse in the sense that, if \\phi(x) \\to \\infty, then there exists a set A with f(n) = r_{A,h}(n) such that card{a in A : |a| \\leq x} < \\phi(x) for all sufficiently large x.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-12-03T19:11:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B34",
      "11B05"
    ],
    "keywords": [
      "representation function",
      "additive basis",
      "asymptotic basis",
      "finite number",
      "necessarily distinct elements"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......2091N"
    }
  }
}