{
  "id": "math/0607209",
  "version": "v4",
  "published": "2006-07-07T19:50:11.000Z",
  "updated": "2006-09-15T17:41:19.000Z",
  "title": "On the Decay of the Fourier Transform and Three Term Arithmetic Progressions",
  "authors": [
    "Ernie Croot"
  ],
  "comment": "One small notational correction: In the paper I called ||f||_(1/3) a `norm', when in fact it should be 'quasinorm'. This does not affect any results, as I don't use the triangle inequality anywhere -- the 1/3 quasinorm was only used as a convenient way to state a corollary of one of my results",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper we prove a basic theorem which says that if f : F_p^n -> [0,1] has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds for quasinorms 1/2-\\delta in place of 1/3), and E(f) = p^{-n} sum_m f(m) is not too ``small'', then there are lots of triples m,m+d,m+2d such that f(m)f(m+d)f(m+2d) > 0. If f is the indicator function for some set S, then this would be saying that the set has many three-term arithmetic progressions. In principle this theorem can be applied to sets having very low density, where |S| is around p^{n(1-c)} for some small c > 0.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2006-09-15T17:41:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P70"
    ],
    "keywords": [
      "fourier transform",
      "three-term arithmetic progressions",
      "indicator function",
      "low density",
      "basic theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7209C"
    }
  }
}