  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.
