Sofic Measures and Densities of Level Sets

The Bernoulli convolution associated to the real β > 1 and the probability vector (p₀, . . . , p_d−1) is a probability measure η_β,p on ℝ, solution of the self-similarity relation η=∑k=0d−1pk⋅η○Sk−1$\eta = \sum\nolimits_{k = 0}^{d - 1} {p_k \cdot \eta \circ S_k^{ - 1} } $, where Sk(x)=x+kβ$S_k (x) = {{x + k} \over \beta }$. If β is an integer or a Pisot algebraic number with finite Rényi expansion, η_β,p is sofic and a Markov chain is naturally associated. If β = b ∈ ℕ and p0=⋯=pd−1=1d$p_0 = \cdots = p_{d - 1} = {1 \over d}$, the study of η_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.