{
  "id": "2112.13759",
  "version": "v2",
  "published": "2021-12-27T16:12:49.000Z",
  "updated": "2023-06-06T16:11:04.000Z",
  "title": "The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches",
  "authors": [
    "Asgar Jamneshan",
    "Terence Tao"
  ],
  "comment": "48 pages, no figures. This is the final version, incorporationg the referee suggestions",
  "categories": [
    "math.CO"
  ],
  "abstract": "We state and prove a quantitative inverse theorem for the Gowers uniformity norm $U^3(G)$ on an arbitrary finite group $G$; the cases when $G$ was of odd order or a vector space over ${\\mathbf F}_2$ had previously been established by Green and the second author and by Samorodnitsky respectively by Fourier-analytic methods, which we also employ here. We also prove a qualitative version of this inverse theorem using a structure theorem of Host--Kra type for ergodic ${\\mathbf Z}^\\omega$-actions of order $2$ on probability spaces established recently by Shalom and the authors.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-06-06T16:11:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "28E05",
      "37A15"
    ],
    "keywords": [
      "arbitrary finite abelian groups",
      "gowers uniformity norm",
      "inverse theorem",
      "ergodic approaches",
      "fourier-analytic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}