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

Existence and exactness of exponential Riesz sequences and frames for fractal measures