Fuglede (1974) conjectured that a domain admits an operator spectrum iff it is possible to tile by a family of translates of . Fuglede proved the conjecture in the special case that the tiling set or the spectrum are lattice subsets of and Iosevich et al. (1999) proved that no smooth symmetric convex body with at least one point of nonvanishing Gaussian curvature can admit an orthogonal basis of exponentials.
Using complex Hadamard matrices of orders 6 and 12, Tao (2003) constructed counterexamples to the conjecture in some small Abelian groups, and lifted these to counterexamples in or .
However, the conjecture has been proved in a great number of special cases (e.g., all convex planar bodies) and remains an open problem in small dimensions. For example, it has been shown in dimension 1 that a nice algebraic characterization of finite sets tiling indeed implies one side of Fuglede's conjecture (Coven and Meyerowitz 1999). Furthermore, it is sufficient to prove these conditions when the tiling gives a factorization of a non-Hajós cyclic group (Amiot 2005).