Tiles with no spectra

2004/06/07 by Mihail N. Kolountzakis, Máté Matolcsi, Kolountzakis, Mihail N. +2 · 5 citations
Computer Science · Mathematics · #20K01 #42B99 #52C22 #Cellular Automata and Applications #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Dynamics and Pattern Formation #math.CA #math.CO #msc:20K01 #msc:42B99 #msc:52C22

paper · pdf · doi:10.48550/arxiv.math/0406127

8 pages

arxiv created 2004/06/07 · openalex publication_date 2004/06/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. 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 ≥ 5) thus disproving one direction of the Spectral Set Conjecture of Fuglede. The other direction was recently disproved by Tao.

Cited by

• Spectra for finite unions of line segments
• Spectrality and non-spectrality of a class of Moran measures with three-element digits
• Spectrality of Prime Size Tiles
• Fuglede's conjecture on orthogonal bases of exponentials
• On the Spectral Properties of a Class of Planar Sierpinski Self-Affine Measures

Related

• Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups
• Tiling sets and spectral sets over finite fields
• Complex Hadamard matrices and the Spectral Set Conjecture
• Both directions of Fuglede's conjecture fail in dimension two
• Packing near the tiling density and exponential bases for product domains