{
  "id": "0902.3093",
  "version": "v1",
  "published": "2009-02-18T10:43:15.000Z",
  "updated": "2009-02-18T10:43:15.000Z",
  "title": "Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis",
  "authors": [
    "Bakir Farhi"
  ],
  "comment": "17 pages",
  "journal": "J. Number Theory, 128 (2008), p. 2214-2230",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $A$ be an additive basis of order $h$ and $X$ be a finite nonempty subset of $A$ such that the set $A \\setminus X$ is still a basis. In this article, we give several upper bounds for the order of $A \\setminus X$ in function of the order $h$ of $A$ and some parameters related to $X$ and $A$. If the parameter in question is the cardinality of $X$, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in $h$ with degree $(|X| + 1)$. Here, by taking instead of the cardinality of $X$ the parameter defined by $d := \\frac{\\diam(X)}{\\gcd\\{x - y | x, y \\in X\\}}$, we show that the order of $A \\setminus X$ is bounded above by $(\\frac{h (h + 3)}{2} + d \\frac{h (h - 1) (h + 4)}{6})$. As a consequence, we deduce that if $X$ is an arithmetic progression of length $\\geq 3$, then the upper bounds of Nathanson and Nash are considerably improved. Further, by considering more complex parameters related to both $X$ and $A$, we get upper bounds which are polynomials in $h$ with degree only 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-18T10:43:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13"
    ],
    "keywords": [
      "upper bounds",
      "additive basis",
      "finite subset",
      "finite nonempty subset",
      "complex parameters"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.3093F"
    }
  }
}