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
$\nu_\lambda$.