Sofic measures and densities of level sets

Abstract

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

arXiv comment31 pages