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).
