  We study Fourier bases on invariant measures generated by affine iterated
function systems in ${\mathbb R}^d$ with integer coefficients. We show that,
for simple digit sets, these systems satisfy the open set condition and have no
overlap. We present natural geometric conditions under which such measures have
an orthonormal basis or a frame of exponential functions with frequencies being
a subset of ${\mathbb Z}^d$. Moreover, we characterize when such measures have
a spectrum in ${\Bbb Z}^d$.
