{
  "id": "0708.2304",
  "version": "v2",
  "published": "2007-08-17T01:15:26.000Z",
  "updated": "2007-08-21T14:57:58.000Z",
  "title": "Inverse problems for linear forms over finite sets of integers",
  "authors": [
    "Melvyn B. Nathanson"
  ],
  "comment": "12 pages; minor corrections",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let f(x_1,x_2,...,x_m) = u_1x_1+u_2 x_2+... + u_mx_m be a linear form with positive integer coefficients, and let N_f(k) = min{|f(A)| : A \\subseteq Z and |A|=k}. A minimizing k-set for f is a set A such that |A|=k and |f(A)| = N_f(k). A finite sequence (u_1, u_2,...,u_m) of positive integers is called complete if {\\sum_{j\\in J} u_j : J \\subseteq {1,2,..,m}} = {0,1,2,..., U}, where $U = \\sum_{j=1}^m u_j.$ It is proved that if f is an m-ary linear form whose coefficient sequence (u_1,...,u_m) is complete, then N_f(k) = Uk-U+1 and the minimizing k-sets are precisely the arithmetic progressions of length k. Other extremal results on linear forms over finite sets of integers are obtained.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-08-21T14:57:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P70",
      "11B25",
      "11B37",
      "11B75",
      "11A25"
    ],
    "keywords": [
      "finite sets",
      "inverse problems",
      "minimizing k-set",
      "m-ary linear form",
      "finite sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.2304N"
    }
  }
}