{
  "id": "1902.01098",
  "version": "v1",
  "published": "2019-02-04T09:37:53.000Z",
  "updated": "2019-02-04T09:37:53.000Z",
  "title": "Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds",
  "authors": [
    "Pablo Candela",
    "Balázs Szegedy"
  ],
  "comment": "35 pages",
  "categories": [
    "math.CO",
    "math.CA",
    "math.NT"
  ],
  "abstract": "We prove a general form of the regularity theorem for uniformity norms, and deduce a generalization of the Green-Tao-Ziegler inverse theorem, extending it to a class of compact nilspaces including all compact abelian groups and nilmanifolds. We derive these results from a structure theorem for cubic couplings, thereby unifying these results with the ergodic structure theorem of Host and Kra. The proofs also involve new results on nilspaces. In particular, we obtain a new stability result for nilspace morphisms. We also strengthen a result of Gutman, Manners and Varju, by proving that a k-step compact nilspace of finite rank is a toral nilspace (in particular, a connected nilmanifold) if and only if its k-dimensional cube set is connected. We also prove that if a morphism from a cyclic group of prime order into a compact finite-rank nilspace is sufficiently balanced (a quantitative form of multidimensional equidistribution), then the nilspace is toral.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-04T09:37:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact abelian groups",
      "uniformity norms",
      "nilmanifold",
      "compact finite-rank nilspace",
      "ergodic structure theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}