{
  "id": "1108.5689",
  "version": "v2",
  "published": "2011-08-29T18:22:09.000Z",
  "updated": "2012-02-21T06:33:53.000Z",
  "title": "Periodicity of the spectrum in dimension one",
  "authors": [
    "Alex Iosevich",
    "Mihail N. Kolountzakis"
  ],
  "comment": "Correction of an error pointed out by Dorin Dutkay; Lemma 1 now has a new proof",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "A bounded measurable set $\\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\\Lambda$ of real numbers (\"frequencies\") such that the exponential functions $e_\\lambda(x) = \\exp(2\\pi i \\lambda x)$, $\\lambda\\in\\Lambda$, form a complete orthonormal system of $L^2(\\Omega)$. Such a set $\\Lambda$ is called a {\\em spectrum} of $\\Omega$. In this note we prove that any spectrum $\\Lambda$ of a bounded measurable set $\\Omega\\subseteq\\RR$ must be periodic.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-02-21T06:33:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodicity",
      "bounded measurable set",
      "complete orthonormal system",
      "lebesgue measure",
      "real line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.5689I"
    }
  }
}