Abstract We consider the self-similar measure _=law (₉ ₀ ⱼ ʲ) on R, where || 1 and the ⱼ are independent, identically distributed with respect to a measure finitely supported on Z. One example of such a measure is a Bernoulli convolution. It is known that for certain combinations of algebraic and uniform on an interval, _ 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_ () is maximal. In this paper, we show that the phenomenon of h_ () being maximal is equivalent to absolute continuity of a self-affine measure _, which is naturally associated to and projects onto _. We also classify all combinations for which this phenomenon occurs: we find that if an algebraic without a Galois conjugate of modulus exactly one has a such that h_ () is maximal, then all Galois conjugates of must be smaller in modulus than one and must satisfy a certain finite set of linear equations in terms of. Lastly, we show that in this case, the measure _ is not only absolutely continuous but also has power Fourier decay, which implies the same for _.
Lauritz Streck (Wed,) studied this question.