A phase transition in random coin tossing

Suppose that a coin with bias theta is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let mu_\theta be the distribution of the observed sequence of coin tosses, and let u_n denote the chance of a renewal at time n. Harris and Keane showed that if sum_{n=1}^infty u_n^2=\infty, then mu_theta and \mu_0 are singular, while if sum_{n=1}^{infty} u_n^2<infty and theta is small enough, then mu_theta is absolutely continuous with respect to mu_0. They conjectured that absolute continuity should not depend on theta, but only on the square-summability of {u_n}. We show that in fact the power law governing the decay of {u_n} is crucial, and for some renewal sequences {u_n}, there is a {phase transition at a critical parameter theta_c in (0,1): for |theta|<theta_c the measures mu_theta$ and mu_0 are mutually absolutely continuous, but for |theta|>theta_c, they are singular. We also prove that when u_n=O(n^{-1}), the measures mu_theta for theta in [-1,1] are all mutually absolutely continuous.