Absolutely continuous measures on the unit circle with sparse Verblunsky coefficients

Orthogonal polynomials and measures on the unit circle are fully determined by their Verblunsky coefficients through the Szegő recurrences. We study measures $\mu$ from the Szegő class whose Verblunsky coefficients vanish off a sequence of positive integers with exponentially growing gaps. All such measures turn out to be absolutely continuous on the circle. We also gather some information about the density function $\mu'$.