Singularity and fine fractal properties of one class of generalized infinite Bernoulli convolutions with essential overlaps. II

Abstract

We discuss the Lebesgue structure and fine fractal properties of infinite Bernoulli convolutions, i.e., the distributions of random variables $\xi=\sum_{k=1}^{\infty}\xi_ka_k$, where $\sum_{k=1}^{\infty}a_k$ is a convergent positive series and $\xi_k$ are independent (generally speaking, nonidentically distributed) Bernoulli random variables. Our main aim is to investigate the class of Bernoulli convolutions with essential overlaps generated by a series $\sum_{k=1}^{\infty}a_k$, such that, for any $k\in \mathbb{N}$, there exists $s_k\in \mathbb{N}\cup\{0\}$ for which $a_k = a_{k+1} = . . . = a_{k+s_k} ≥ r_{k+s_k}$ and, in addition, $s_k > 0$ for infinitely many indices $k$. In this case, almost all (both in a sense of Lebesgue measure and in a sense of fractal dimension) points from the spectrum have continuum many representations of the form $\xi=\sum_{k=1}^{\infty}\varepsilon_ka_k$, with $\varepsilon_k\in\{0, 1\}$. It is proved that $\mu_\xi$ has either a pure discrete distribution or a pure singulary continuous distribution.
We also establish sufficient conditions for the faithfulness of the family of cylindrical intervals on the spectrum $\mu_\xi$ generated by the distributions of the random variables $\xi$. In the case of singularity, we also deduce the explicit formula for the Hausdorff dimension of the corresponding probability measure [i.e., the Hausdorff–Besicovitch dimension of the minimal supports of the measure $\mu_\xi$ (in a sense of dimension)].