  We study the construction of exponential frames and Riesz sequences for a
class of fractal measures on ${\mathbb R}^d$ generated by infinite convolution
of discrete measures using the idea of frame towers and Riesz-sequence towers.
The exactness and overcompleteness of the constructed exponential frame or
Riesz sequence is completely classified in terms of the cardinality at each
level of the tower. Using a version of the solution of the Kadison-Singer
problem, known as the $R_{\epsilon}$-conjecture, we show that all these
measures contain exponential Riesz sequences of infinite cardinality.
Furthermore, when the measure is the middle-third Cantor measure, or more
generally for self-similar measures with no-overlap condition, there are always
exponential Riesz sequences of maximal possible Beurling dimension.
