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

In this paper, we consider the self-similar measure ${\nu }_{\lambda }=\mathrm{l}\mathrm{a}\mathrm{w}\left({\sum }_{j\ge 0}{\xi }_j{\lambda }^j\right)$ on $\mathbb{R}$, where $\mathrm{\mid }\lambda \mathrm{\mid }<1$ and the ${\xi }_j\sim \nu$ are independent, identically distributed with respect to a measure $\nu$ finitely supported on $\mathbb{Z}$. One example of this is the classical 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 (\cite{garsia1}, \cite{feng}); in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy $h_{\lambda }(\nu )$ is maximal. We show that absolute continuity and power Fourier decay occur when $\lambda$ and $\nu$ are such that $h_{\lambda }(\nu )$ is maximal and classify all combinations for which this is the case. 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 $\lambda$.