Spectral pairs in Cartesian coordinates

Let $ \Omega \subset R^d $ have finite positive Lebesgue measure, and let $ \mathcal{L}^{2}(\Omega) $ be the corresponding Hilbert space of $ \mathcal{L}^{2} $-functions on $ \Omega $. We shall consider the exponential functions $ e_{\lambda} $ on $ \Omega $ given by $ e_{\lambda}(x)=e^{i2\pi\lambda x} $. If these functions form an orthogonal basis for $ \mathcal{L}^{2}(\Omega) $, when $ \lambda $ ranges over some subset $ \Lambda $ in $ R^d $, then we say that $ (\Omega,\Lambda) $ is a spectral pair, and that $ \Lambda $ is a spectrum. We conjecture that $ (\Omega,\Lambda) $ is a spectral pair if and only if the translates of some set $ \Omega' $ by the vectors of $ \Lambda $ tile $ R^d $. In the special case of $ \Omega=I^d $, the $ d $-dimensional unit cube, we prove this conjecture, with $ \Omega'=I^d $, for $ d \leq 3 $, describing all the tilings by $ I^d $, and for all $ d $ when $ \Lambda $ 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.