{
  "id": "1609.06489",
  "version": "v1",
  "published": "2016-09-21T10:25:45.000Z",
  "updated": "2016-09-21T10:25:45.000Z",
  "title": "An application of the sum-product phenomenon to sets having no solutions of several linear equations",
  "authors": [
    "Ilya D. Shkredov"
  ],
  "comment": "31 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We prove that for an arbitrary $\\kappa < \\frac{5}{31}$ any subset of $\\mathbf{F}_p$ avoiding $t$ linear equations with three variables has size less than $O(p/t^\\kappa)$. We also find several applications to problems about so--called non--averaging sets, number of collinear triples and mixed energies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-21T10:25:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear equations",
      "sum-product phenomenon",
      "application",
      "collinear triples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}