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.