{
  "id": "0901.2602",
  "version": "v2",
  "published": "2009-01-17T01:11:01.000Z",
  "updated": "2009-06-08T04:16:35.000Z",
  "title": "An inverse theorem for the uniformity seminorms associated with the action of $F^ω$",
  "authors": [
    "Vitaly Bergelson",
    "Terence Tao",
    "Tamar Ziegler"
  ],
  "comment": "59 pages, 2 figures, to appear, GAFA. Referee suggestions incorporated. Also, the paper has been shortened at the request of the journal; previous versions on the arXiv can thus be viewed as extended versions",
  "journal": "Geom. Funct. Anal. 19 (2010), No. 6, 1539-1596",
  "doi": "10.1007/s00039-010-0051-1",
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "Let $\\F$ a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm $U^k(\\X)$ for an ergodic action $(T_g)_{g \\in \\F^\\omega}$ of the infinite abelian group $\\F^\\omega$ on a probability space $X = (X,\\B,\\mu)$ is generated by phase polynomials $\\phi: X \\to S^1$ of degree less than $C(k)$ on $X$, where $C(k)$ depends only on $k$. In the case where $k \\leq \\charac(\\F)$ we obtain the sharp result $C(k)=k$. This is a finite field counterpart of an analogous result for $\\Z$ by Host and Kra. In a companion paper to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case $k \\leq \\charac(\\F)$, with a partial result in low characteristic.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-06-08T04:16:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A35"
    ],
    "keywords": [
      "uniformity seminorms",
      "inverse theorem",
      "high characteristic case",
      "gowers-host-kra uniformity seminorm",
      "infinite abelian group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 59,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0901.2602B"
    }
  }
}