Fuglede's conjecture is an open problem in mathematics proposed by Bent Fuglede in 1974.
It states that every domain of
(i.e. subset of
with positive finite Lebesgue measure) is a spectral set if and only if it tiles
by translation.
[1] Spectral sets in
with positive finite Lebesgue measure is said to be a spectral set if there exists a
2 π i
λ , ⋅
λ ∈
is an orthogonal basis of
is then said to be a spectrum of
is called a spectral pair.
Translational tiles of
by translation (i.e.
is a translational tile) if there exist a discrete set
and the Lebesgue measure of