{
  "id": "2307.07213",
  "version": "v1",
  "published": "2023-07-14T08:15:25.000Z",
  "updated": "2023-07-14T08:15:25.000Z",
  "title": "On the maximal spectral type of nilsystems",
  "authors": [
    "Ethan Ackelsberg",
    "Florian K. Richter",
    "Or Shalom"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(G/\\Gamma,R_a)$ be an ergodic $k$-step nilsystem for $k\\geq 2$. We adapt an argument of Parry to show that $L^2(G/\\Gamma)$ decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host, Kra and Maass for $2$-step nilsystems and a result by Stepin for nilsystems $G/\\Gamma$ with connected, simply connected $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-14T08:15:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximal spectral type",
      "step nilsystem",
      "discrete spectrum",
      "lebesgue spectrum",
      "infinite multiplicity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}