{
  "id": "0810.5527",
  "version": "v2",
  "published": "2008-10-30T17:10:41.000Z",
  "updated": "2009-10-29T17:56:38.000Z",
  "title": "The inverse conjecture for the Gowers norm over finite fields via the correspondence principle",
  "authors": [
    "Terence Tao",
    "Tamar Ziegler"
  ],
  "comment": "21 pages, no figures, to appear, Analysis & PDE. This is the final version, incorporating the referee's corrections",
  "journal": "Analysis & PDE Vol. 3 (2010), No. 1, 1-20",
  "doi": "10.2140/apde.2010.3.1",
  "categories": [
    "math.CO",
    "math.DS"
  ],
  "abstract": "The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\\|f\\|_{U^d(V)}$ if and only if it correlates with a phase polynomial $\\phi = e_\\F(P)$ of degree at most $d-1$, thus $P: V \\to \\F$ is a polynomial of degree at most $d-1$. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case $\\charac(F) \\geq d$ from an ergodic theory counterpart, which was recently established by Bergelson and the authors. In low characteristic we obtain a partial result, in which the phase polynomial $\\phi$ is allowed to be of some larger degree $C(d)$. The full inverse conjecture remains open in low characteristic; the counterexamples by Lovett-Meshulam-Samorodnitsky or Green-Tao in this setting can be avoided by a slight reformulation of the conjecture.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-10-29T17:56:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06",
      "37A15"
    ],
    "keywords": [
      "finite field",
      "correspondence principle",
      "full inverse conjecture remains open",
      "low characteristic",
      "phase polynomial"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.5527T"
    }
  }
}