Tiles with no spectra
-
Mihail N Kolountzakis
and
Máté Matolcsi
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 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 ℤd and ℝd (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].
© Walter de Gruyter
Articles in the same Issue
- Measurable metrics and Gaussian concentration
- Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids
- Spherical means on compact locally symmetric spaces of non-positive curvature
- Reflections acting efficiently on a building
- Decay rates of oscillatory integral operators in “1 + 2” dimensions
- Singular perturbation for the first eigenfunction and blow-up analysis
- Tiles with no spectra
- Pointwise estimates for a model problem in nonlinear elasticity
Articles in the same Issue
- Measurable metrics and Gaussian concentration
- Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids
- Spherical means on compact locally symmetric spaces of non-positive curvature
- Reflections acting efficiently on a building
- Decay rates of oscillatory integral operators in “1 + 2” dimensions
- Singular perturbation for the first eigenfunction and blow-up analysis
- Tiles with no spectra
- Pointwise estimates for a model problem in nonlinear elasticity
