{
  "id": "1604.08940",
  "version": "v1",
  "published": "2016-04-29T19:12:46.000Z",
  "updated": "2016-04-29T19:12:46.000Z",
  "title": "Comparison estimates for linear forms in additive number theory",
  "authors": [
    "Melvyn B. Nathanson"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $M$ be an $R$-module. Consider the $h$-ary linear form $\\Phi:M^h \\rightarrow M$ defined by $\\Phi(t_1,\\ldots, t_h) = \\sum_{i=1}^h \\varphi_it_i$with nonzero coefficient sequence $(\\varphi_1,\\ldots, \\varphi_h) \\in R^h$. For every subset $A$ of $M$, define \\[ \\Phi(A) = \\{ \\Phi(a_1,\\ldots, a_h) : (a_1,\\ldots, a_h) \\in A^h \\}. \\] For every nonempty subset $I$ of $\\{1,2,\\ldots, h\\}$, define the subsequence sum $s_I = \\sum_{i\\in I} \\varphi_i$. Let $\\mathcal{S}(\\Phi) = \\{s_I: I \\subseteq \\{1,2,\\ldots, h\\}, I \\neq \\emptyset \\} $ be the set of all nonempty subsequence sums of the sequence of coefficients of $\\Phi$. Let $R^{\\times}$ be the group of units of the ring $R$. Theorem. Let $\\Upsilon(t_1,\\ldots, t_g) = \\sum_{i=1}^g \\upsilon_it_i$ and $\\Phi(t_1,\\ldots, t_h) = \\sum_{i=1}^h \\varphi_it_i$ be linear forms with nonzero coefficients in the ring $R$. If $\\{0, 1\\} \\subseteq \\mathcal{S}(\\Upsilon)$ and $\\mathcal{S}(\\Phi) \\subseteq R^{\\times}$, then for every $\\varepsilon > 0$ and $c > 1$ there exists a finite $R$-module $M$ with $|M| > c$ and a subset $A$ of $M$ such that $\\Upsilon(A \\cup \\{0\\}) = M$ and $|\\Phi(A)| < \\varepsilon |M|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-29T19:12:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11B13",
      "11B30",
      "11B75",
      "11P99"
    ],
    "keywords": [
      "additive number theory",
      "comparison estimates",
      "ary linear form",
      "nonzero coefficient sequence",
      "nonempty subsequence sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160408940N"
    }
  }
}