On the distribution of a random power series on the dyadic half-line

We consider an analog of the problem of the existence of the summable distributional density of a random variable in the form of power series on the dyadic half-line which was originally proposed and partially solved by Erdös on the standard real line. Given a random variable $\xi$ as a series of the powers of $\lambda \in (0, 1)$, we address the question of $\lambda$ such that the density of $\xi$ belongs to the space of the function whose modulus is summable on the dyadic half-line. We answer the question for some values of $\lambda$, and consider the so-called dual problem when $\lambda = \frac{1}{2}$ is fixed, but the coefficients of the formula for $\xi$ have more degrees of freedom. Also we obtain some criteria for the existence of density in terms of the solution of the refinement equation tied directly to $\xi$ as well as in terms of the coefficients defining $\xi$.