Both directions of Fuglede's conjecture fail in dimension two

2026/07/17 by Tao Zhang
#math.CO #math.CA

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Abstract

Fuglede's conjecture asserts that a measurable set of positive and finite measure is spectral if and only if it tiles Euclidean space by translations. Counterexamples are known in every dimension d≥3, whereas the one- and two-dimensional cases have remained unresolved. We construct two explicit 60-point subsets of the rank-two finite Abelian group \Z₆₀×\Z₁₂: one is a translational tile with no spectrum, and the other is spectral but does not tile. A finite-to-infinite transference principle lifts them to bounded subsets of \R² that are finite unions of unit squares. Consequently, both implications in Fuglede's conjecture fail in dimension two.

Citations

• The Fuglede conjecture for convex domains is true in all dimensions

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