Sofic measures and densities of level sets

The Bernoulli convolution associated to the real $\beta>1$ and the probability vector $(p_0,\dots,p_{d-1})$ is a probability measure $\eta_{\beta,p}$ on $\mathbb R$, solution of~the self-similarity relation $\eta=\sum_{k=0}^{d-1}p_k\cdot\eta\circ S_k^{-1}$, where $S_k(x)=\frac{x+k}\beta$. If $\beta$ is an integer or a Pisot algebraic number with finite R\'enyi expansion, $\eta_{\beta,p}$ is sofic and a Markov chain is naturally associated. If $\beta=b\in\mathbb N$ and $p_0=\dots=p_{d-1}=\frac1d$, the study of $\eta_{b,p}$ is close to the study of the order of growth of the number of~representations in base $b$ with digits in $\{0,1,\dots,d-1\}$. In the case $b=2$ and $d=3$ it has also something to do with the metric properties of~the continued fractions.