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

• Date: 2015
• Main Authors: Lebid', M. V., Torbin, H. M., Лебідь, М. В., Торбін, Г. М.
• Format: Article
• Language: Ukrainian
• Published: Institute of Mathematics, NAS of Ukraine 2015
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/2099
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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)].