Tiles with no spectra

Kolountzakis, M. N. and Matolcsi, Máté (2006) Tiles with no spectra. FORUM MATHEMATICUM, 18 (3). pp. 519-528. ISSN 0933-7741

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Abstract

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 L-2 space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang [7]. 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 [1]. The other direction was recently disproved by Tao [12].

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