{
  "id": "math/0607208",
  "version": "v1",
  "published": "2006-07-07T19:43:19.000Z",
  "updated": "2006-07-07T19:43:19.000Z",
  "title": "On the Structure of Sets with Few Three-Term Arithmetic Progressions",
  "authors": [
    "Ernie Croot"
  ],
  "comment": "This is a much cleaner version of a proof published on the arxives three years ago, but where this one holds for finite fields F_p^n. The result in this paper is much clearer than that published previously",
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix a density d in (0,1], and let F_p^n be a finite field, where we think of p fixed and n tending to infinity. Let S be any subset of F_p^n having the minimal number of three-term progressions, subject to the constraint |S| is at least dp^n. We show that S must have some structure, and that up to o(p^n) elements, it is a union of a small number of cosets of a subspace of dimension n-o(n).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-07-07T19:43:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P70"
    ],
    "keywords": [
      "three-term arithmetic progressions",
      "finite field",
      "dimension n-o",
      "minimal number",
      "three-term progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7208C"
    }
  }
}