{
  "id": "1101.1469",
  "version": "v2",
  "published": "2011-01-07T16:45:30.000Z",
  "updated": "2011-09-08T14:47:15.000Z",
  "title": "The inverse conjecture for the Gowers norm over finite fields in low characteristic",
  "authors": [
    "Terence Tao",
    "Tamar Ziegler"
  ],
  "comment": "68 pages, no figures, to appear, Annals of Combinatorics. This is the final version, incorporating the referee's suggestions",
  "categories": [
    "math.CO"
  ],
  "abstract": "We establish the \\emph{inverse conjecture for the Gowers norm over finite fields}, which asserts (roughly speaking) that if a bounded function $f: V \\to \\C$ on a finite-dimensional vector space $V$ over a finite field $\\F$ has large Gowers uniformity norm $\\|f\\|_{U^{s+1}(V)}$, then there exists a (non-classical) polynomial $P: V \\to \\T$ of degree at most $s$ such that $f$ correlates with the phase $e(P) = e^{2\\pi i P}$. This conjecture had already been established in the \"high characteristic case\", when the characteristic of $\\F$ is at least as large as $s$. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson, together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work of Green and the first author and of Kaufman and Lovett.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-09-08T14:47:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "11T06"
    ],
    "keywords": [
      "finite field",
      "gowers norm",
      "low characteristic",
      "large gowers uniformity norm",
      "inverse conjecture established earlier"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 68,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.1469T"
    }
  }
}