{
  "id": "1002.2209",
  "version": "v1",
  "published": "2010-02-10T20:51:19.000Z",
  "updated": "2010-02-10T20:51:19.000Z",
  "title": "Linear forms and quadratic uniformity for functions on $\\mathbb{F}_p^n$",
  "authors": [
    "W. T. Gowers",
    "J. Wolf"
  ],
  "comment": "26 pages",
  "doi": "10.1112/S0025579311001264",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We give improved bounds for our theorem in [GW09], which shows that a system of linear forms on $\\mathbb{F}_p^n$ with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of $\\mathbb{F}_p^n$. While in [GW09] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [GrT08], we use the Hahn-Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the $U^3$ inverse theorem [GrT08].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-10T20:51:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30"
    ],
    "keywords": [
      "linear forms",
      "quadratic uniformity",
      "quadratically uniform part",
      "hahn-banach theorem",
      "structure theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.2209G"
    }
  }
}