{
  "id": "1908.02794",
  "version": "v1",
  "published": "2019-08-07T18:43:14.000Z",
  "updated": "2019-08-07T18:43:14.000Z",
  "title": "Spectrum is rational in dimension one",
  "authors": [
    "Chun-Kit Lai",
    "Yang Wang"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "A bounded measurable set $\\Omega\\subset{\\mathbb R}^d$ is called a spectral set if it admits some exponential orthonormal basis $\\{e^{2\\pi i \\langle\\lambda,x\\rangle}: \\lambda\\in\\Lambda\\}$ for $L^2(\\Omega)$. In this paper, we show that in dimension one $d=1$, any spectrum $\\Lambda$ with $0\\in\\Lambda$ of a spectral set $\\Omega$ with Lebesgue measure normalized to 1 must be rational. Combining previous results that spectrum must be periodic, the Fuglede's conjecture on ${\\mathbb R}^1$ is now equivalent to the corresponding conjecture on all cyclic groups ${\\mathbb Z}_{n}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-07T18:43:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral set",
      "exponential orthonormal basis",
      "lebesgue measure",
      "fugledes conjecture",
      "cyclic groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}