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arXiv · 2607.15632

Both directions of Fuglede's conjecture fail in dimension two

Tao Zhang
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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 $d\ge3$, whereas the one- and two-dimensional cases have remained unresolved. We construct two explicit $60$-point subsets of the rank-two finite Abelian group $\Z_{60}\times\Z_{12}$: 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 $\R^2$ that are finite unions of unit squares. Consequently, both implications in Fuglede's conjecture fail in dimension two.

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Tao Zhang. 2026-07-17. Both directions of Fuglede's conjecture fail in dimension two. https://arxiv.org/abs/2607.15632

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