On absolute continuity and maximal Garsia entropy for self-similar measures with algebraic contraction ratio

Abstract We consider the self-similar measure $\nu_\lambda=\text{law}\left(\sum_{j \geq 0} \xi_j \lambda^j\right)$ on $\mathbb{R}$ , where $|\lambda| \lt 1$ and the $\xi_j \sim \nu$ are independent, identically distributed with respect to a measure $\nu$ finitely supported on $\mathbb{Z}$ . One example of such a measure is a Bernoulli convolution. It is known that for certain combinations of algebraic $\lambda$ and $\nu$ uniform on an interval, $\nu_\lambda$ is absolutely continuous and its Fourier transform has power decay; in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy $h_{\lambda}(\nu)$ is maximal. In this paper, we show that the phenomenon of $h_{\lambda}(\nu)$ being maximal is equivalent to absolute continuity of a self-affine measure $\mu_\lambda$ , which is naturally associated to $\lambda$ and projects onto $\nu_\lambda$ . We also classify all combinations for which this phenomenon occurs: we find that if an algebraic $\lambda$ without a Galois conjugate of modulus exactly one has a $\nu$ such that $h_{\lambda}(\nu)$ is maximal, then all Galois conjugates of $\lambda$ must be smaller in modulus than one and $\nu$ must satisfy a certain finite set of linear equations in terms of $\lambda$ . Lastly, we show that in this case, the measure $\mu_\lambda$ is not only absolutely continuous but also has power Fourier decay, which implies the same for <jats:alternatives