THE DISPROVAL OF FUGLEDE’S CONJECTURE

MÁTÉ MATOLCSI

Abstract. Fuglede’s conjecture relates the notions of translational tiles and spectral sets in ℝ^{d}.

Let Ω ⊂ ℝ^{d} be a Lebesgue measureable set of finite non-zero measure. A set Λ ⊂ ℝ^{d} is said to be a spectrum of Ω if the functions \{\frac{1}{|Ω|^{1/2}} e^{2πi⟨x,ξ⟩}\}_{ξ∈Λ} form an orthonormal basis of L^{2}(Ω) (here |Ω| denotes the Lebesgue measure of Ω). If such Λ exists, Ω is said to be spectral. The set Ω is said to be a (translational) tile if it is possible to tile ℝ^{d} with a family of translates {t + Ω : t ∈ Σ} of Ω (ignoring overlaps and gaps of measure zero).

Then, Fuglede formulated the following

Conjecture 0.1. [2]A set Ω ⊂ ℝ^{d} of finite, non-zero Lebesgue measure is a tile if and only if it is spectral.

Fuglede [2] proved the conjecture in the special case when the spectrum Λ or the translation set Σ is assumed to be a lattice. The general case of the conjecture was open for nearly 30 years, until last year Tao [1] showed an example to disprove one direction of the conjecture in 5 and higher dimensions. Namely, he gave an example of a spectral set in ℝ^{5} which is not a tile.

With a slight modification of Tao’s arguments I reduced the dimension of the counterexample to 4, and subsequently (in a joint work with Prof. M. Kolountzakis) to 3.

The ’tile → spectral’ direction, however, could not be tackled by Tao’s arguments. A counterexample for this direction was found in dimension 5 during my visit to Prof. M. Kolountzakis at the University of Crete node of the HARP Network. The proof is mainly based on Fourier analysis and combinatorics. Recently, in a joint work Szilard Revesz and Balint Farkas reduced this dimension to 4, and subsequently Balint Farkas and I reduced the dimension furhter to 3.

At present the Conjecture is still open in dimensions 1 and 2. In dimension 1, the ’tile → spectral’ part of the conjecture would follow from a particular number theoretic conjecture (which is far from being settled...)

References

[1] T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Letters, to appear.

[2] B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16(1974), 101-121.

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