Both directions of Fuglede's conjecture fail in dimension two

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.