{
  "id": "1403.7040",
  "version": "v2",
  "published": "2014-03-27T14:12:46.000Z",
  "updated": "2014-05-16T19:55:44.000Z",
  "title": "On systems of complexity one in the primes",
  "authors": [
    "Kevin Henriot"
  ],
  "comment": "33 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Consider a translation-invariant system of linear equations $V x = 0$ of complexity one, where $V$ is an integer $r \\times t$ matrix. We show that if $A$ is a subset of the primes up to $N$ of density at least $C(\\log\\log N)^{-1/25t}$, there exists a solution $x \\in A^t$ to $V x = 0$ with distinct coordinates. This extends a quantitative result of Helfgott and de Roton for three-term arithmetic progressions, while the qualitative result is known to hold for all systems of equations of finite complexity by the work of Green and Tao.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-16T19:55:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "three-term arithmetic progressions",
      "linear equations",
      "distinct coordinates",
      "translation-invariant system",
      "finite complexity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7040H"
    }
  }
}