{
  "id": "math/0612016",
  "version": "v1",
  "published": "2006-12-01T10:50:46.000Z",
  "updated": "2006-12-01T10:50:46.000Z",
  "title": "On Fuglede's conjecture and the existence of universal spectra",
  "authors": [
    "Bálint Farkas",
    "Máté Matolcsi",
    "Péter Móra"
  ],
  "comment": "12 pages",
  "journal": "Journal of Fourier Analysis and Applications, Volume 12, Number 5 / October, 2006, 483-494",
  "doi": "10.1007/s00041-005-5069-7",
  "categories": [
    "math.CA"
  ],
  "abstract": "Recent methods developed by Tao \\cite{tao}, Kolountzakis and Matolcsi \\cite{nspec} have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in $\\RR^5$ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a non-spectral tile. In search of lower dimensional non-spectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang \\cite{lagwang}. In particular, we prove here that the USC and the ``tile $\\to$ spectral'' direction of Fuglede's conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szab\\'o \\cite{lagszab} for the existence of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples of tiles which have no spectra. We overcome these difficulties by invoking some ideas of R\\'ev\\'esz and Farkas \\cite{revfark}, and obtain non-spectral tiles in $\\RR^3$. Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The 1 dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz \\cite{covmey}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-12-01T10:50:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C22",
      "20K01",
      "42B99"
    ],
    "keywords": [
      "fugledes conjecture",
      "universal spectrum conjecture remains open",
      "lower dimensional non-spectral tiles",
      "fugeldes spectral set conjecture"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12016F"
    }
  }
}