Spectral pairs in Cartesian coordinates
Let $ Ω\subset R^d $ have finite positive Lebesgue measure, and let $ \mathcal{L}^{2}(Ω) $ be the corresponding Hilbert space of $ \mathcal{L}^{2} $-functions on $ Ω$. We shall consider the exponential functions $ e_λ $ on $ Ω$ given by $ e_λ(x)=e^{i2πλx} $. If these functions form an orthogonal basis for $ \mathcal{L}^{2}(Ω) $, when $ λ$ ranges over some subset $ Λ$ in $ R^d $, then we say that $ (Ω,Λ) $ is a spectral pair, and that $ Λ$ is a spectrum. We conjecture that $ (Ω,Λ) $ is a spectral pair if and only if the translates of some set $ Ω' $ by the vectors of $ Λ$ tile $ R^d $. In the special case of $ Ω=I^d $, the $ d $-dimensional unit cube, we prove this conjecture, with $ Ω'=I^d $, for $ d \leq 3 $, describing all the tilings by $ I^d $, and for all $ d $ when $ Λ$ is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.