{
  "id": "1801.07135",
  "version": "v1",
  "published": "2018-01-22T15:24:11.000Z",
  "updated": "2018-01-22T15:24:11.000Z",
  "title": "Maximising the number of solutions to a linear equation in a set of integers",
  "authors": [
    "James Aaronson"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Given a linear equation of the form $a_1x_1 + a_2x_2 + a_3x_3 = 0$ with integer coefficients $a_i$, we are interested in maximising the number of solutions to this equation in a set $S \\subseteq \\mathbb{Z}$, for sets $S$ of a given size. We prove that, for any choice of constants $a_1, a_2$ and $a_3$, the maximum number of solutions is at least $(\\frac{1}{12} + o(1))|S|^2$. Furthermore, we show that this is optimal, in the following sense. For any $\\varepsilon > 0,$ there are choices of $a_1, a_2$ and $a_3,$ for which any large set $S$ of integers has at most $(\\frac{1}{12} + \\varepsilon)|S|^2$ solutions. For equations in $k \\geq 3$ variables, we also show an analogous result. Define $\\sigma_k = \\int_{-\\infty}^{\\infty} \\frac{\\sin \\pi x}{\\pi x} dx.$ Then, for any choice of constants $a_1, \\dots, a_k$, there are sets $S$ with at least $(\\frac{\\sigma_k}{k^{k-1}} + o(1))|S|^{k-1}$ solutions to $a_1x_1 + \\dots + a_kx_k = 0$. Moreover, there are choices of coefficients $a_1, \\dots, a_k$ for which any large set $S$ must have no more than $(\\frac{\\sigma_k}{k^{k-1}} + \\varepsilon)|S|^{k-1}$ solutions, for any $\\varepsilon > 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-22T15:24:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear equation",
      "large set",
      "maximising",
      "maximum number",
      "integer coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}