Spectral Sets Whose Spectrum Is a Lattice with a Base

LetLbe a finite subset of Euclideand-dimensional space Rdand letSbe a lattice in Rdsuch that {l−l′ : l, l′∈L}∩S={0}. A classification of all Lebesgue measurable subsetsΩof Rdfor which the functionseλ(x)=exp(i2πλx),x∈Ω,λ∈L+S={l+s : l∈L, s∈S} form an orthogonal basis for L2(Ω) is given in terms of geometric and combinatorial data; the data is derived fromLandS.