{
  "id": "1408.2568",
  "version": "v1",
  "published": "2014-08-11T21:55:20.000Z",
  "updated": "2014-08-11T21:55:20.000Z",
  "title": "Roth's theorem for four variables and additive structures in sums of sparse sets",
  "authors": [
    "Tomasz Schoen",
    "Olof Sisask"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A+A+A contain long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional subspaces if A is a subset of a vector space over a finite field, even if A has density of the shape above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-11T21:55:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "11B25"
    ],
    "keywords": [
      "sparse sets",
      "roths theorem",
      "additive structures",
      "contain long arithmetic progressions",
      "vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2568S"
    }
  }
}