Fuglede's conjecture


Fuglede's conjecture is an open problem in mathematics proposed by Bent Fuglede in 1974. It states that every domain of is a spectral set if and only if it tiles by translation.

Spectral sets and translational tiles

Spectral sets in
A set with positive finite Lebesgue measure is said to be a spectral set if there exists a such that is an orthogonal basis of. The set is then said to be a spectrum of and is called a spectral pair.
Translational tiles of
A set is said to tile by translation if there exist a discrete set such that and the Lebesgue measure of is zero for all in.

Partial results