Self-affine spectral measures and frame spectral measures on ${\mathbb R}^d$

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 ${\Bbb Z}^d$.