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Mihail N Kolountzakis, Mt Matolcsi

Tiles with no spectra

We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their L² space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang [Lagarias J. C. and Wang Y.: Spectral sets and factorizations of finite Abelian groups.J. Func. Anal. 145 (1997), 73–98]. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups ℤ^d and ℝ^d (for d ≥5 ) thus disproving one direction of the Spectral Set Conjecture of Fuglede [Fuglede B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16 (1974), 101–121]. The other direction was recently disproved by Tao [Tao T.: Fuglede's conjecture is false in 5 and higher dimensions. Math. Res. Letters 11 (2004), 251–258].

Forum Mathematicum, Walter de Gruyter

Print ISSN: 0933-7741
Volume: 18, 05/2006
Pages: 519 - 528

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