The Bernoulli convolution associated to the real \( \beta > 1 \) and the probability vector \( (p_0, \ldots, p_{d-1}) \) 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 \( S_k(x) = \dfrac{x+k}{\beta} \). 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 \( p_0 = \cdots = p_{d-1} = \dfrac{1}{d} \), 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 \( \{0, 1, \ldots, d-1\} \). In the case \( b = 2 \) and \( d = 3 \) it has something to do with the metric properties of the continued fractions.