Both directions of Fuglede's conjecture fail in dimension two

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 $d\ge 3$, whereas the one- and two-dimensional cases have remained unresolved. We construct two explicit $60$-point subsets of the rank-two finite Abelian group ${\mathbb{Z}}_{60}\times {\mathbb{Z}}_{12}$: 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 ${\mathbb{R}}^2$ that are finite unions of unit squares. Consequently, both implications in Fuglede's conjecture fail in dimension two.

arXiv comment18 pages