[Paper Review] Tiles with no spectra

Mihail N. Kolountzakis, Máté Matolcsi|ArXiv.org|Jun 7, 2004

Mathematical Analysis and Transform Methods4 citations

TL;DR

This paper disproves the 'tiling implies spectral' direction of Fuglede's Spectral Set Conjecture in dimensions d ≥ 5 by constructing a finite subset of a finite Abelian group that tiles but lacks a common spectrum across its tiling complements. Using a counterexample to the Universal Spectrum Conjecture of Lagarias and Wang, the authors extend this to non-spectral tiles in ℤ^d and ℝ^d, demonstrating that tiling does not imply spectral property in higher dimensions.

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. 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.

Motivation & Objective

• To disprove the 'tiling implies spectral' direction of Fuglede's Spectral Set Conjecture in dimensions d ≥ 5.
• To construct a finite set in a finite Abelian group that tiles but has no spectrum, thereby refuting the Universal Spectrum Conjecture of Lagarias and Wang.
• To extend non-spectral tiles from finite groups to ℤ^d and ℝ^d using lifting theorems, demonstrating that tiling does not imply spectral property in Euclidean spaces of dimension ≥ 5.
• To provide a counterexample to the conjecture that every tiling set in ℝ^d is spectral, completing the disproof of Fuglede's conjecture in both directions.

Proposed method

• Construct a counterexample to the Universal Spectrum Conjecture in the finite group ℤ₆⁵ × ℤ₁₅ by identifying a set A that tiles but whose tiling complements lack a common spectrum.
• Use the structure of the Fourier transform of the indicator function χ_A to show that no orthogonal basis of characters (spectrum) exists for A in the finite group setting.
• Lift the non-spectral tile from the finite group ℤ₆⁵ × ℤ₁₅ to ℤ⁶ via a construction in Theorem 4.1, preserving tiling and non-spectral properties.
• Apply Theorem 4.2 to lift the non-spectral tile from ℤ^d to ℝ^d by forming E = A + [0,1)^d, showing that E tiles ℝ^d but has no spectrum.
• Use the equivalence between spectral sets in ℤ^d and ℝ^d established in Theorem 4.2 to transfer spectral properties between the discrete and continuous settings.
• Employ the condition that a set Λ is a spectrum for Ω if and only if ∑_{λ∈Λ} |χ̂_Ω|²(x−λ) = |Ω|² a.e. on the dual group, to verify spectral and non-spectral properties via Fourier zero sets.

Experimental results

Research questions

• RQ1 Does every set that tiles ℝ^d by translation necessarily have a spectrum (i.e., an orthogonal basis of characters in L²(Ω))?
• RQ2 Can a set in a finite Abelian group tile the group yet fail to have a common spectrum among all its tiling complements?
• RQ3 Is the Universal Spectrum Conjecture of Lagarias and Wang valid for finite Abelian groups?
• RQ4 Can non-spectral tiles in finite groups be lifted to non-spectral tiles in ℤ^d and ℝ^d?
• RQ5 Does the 'tiling implies spectral' direction of Fuglede's Spectral Set Conjecture hold in dimensions d ≥ 5?

Key findings

• A finite subset of the group ℤ₆⁵ × ℤ₁₅ is constructed that tiles the group but whose tiling complements do not share a common spectrum, disproving the Universal Spectrum Conjecture of Lagarias and Wang.
• The constructed set in ℤ₆⁵ × ℤ₁₅ is a non-spectral tile, meaning it tiles but has no spectrum in the finite group setting.
• This non-spectral tile is lifted to ℤ^6 using Theorem 4.1, resulting in a set that tiles ℤ^6 but is not spectral.
• By Theorem 4.2, the set A + [0,1)^6 in ℝ^6 is shown to tile ℝ^6 but is not spectral, establishing a counterexample in ℝ^6.
• The construction is adapted to ℤ₆⁴ × ℤ₁₀₂ (since 6 and 17 are coprime), yielding a non-spectral tile in dimension 5, which is then extended to ℝ^5.
• For all d ≥ 5, there exist sets in ℝ^d that tile ℝ^d by translation but are not spectral, thus disproving the 'tiling implies spectral' direction of Fuglede's Spectral Set Conjecture.