Tiles with no spectra

Details Tiles with no spectra Tiles with no spectra

2004-06-07

arxiv.org/abs/math/0406127v1

Mathematics

Classical Analysis and ODEs

8 pages

Scientific paper

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. 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 $\ZZ^d$ and $\RR^d$ (for $d \ge 5$) thus disproving one direction of the Spectral Set Conjecture of Fuglede. The other direction was recently disproved by Tao.

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