Singularity and fine fractal properties of one class of generalized infinite Bernoulli convolutions with essential overlaps. II - Lebid', M. V.; Torbin, H. M.; Лебідь, М. В.; Торбін, Г. М. (Institute of Mathematics, NAS of Ukraine, 2015)
We discuss the Lebesgue structure and fine fractal properties of infinite Bernoulli convolutions, i.e., the distributions of random variables $ξ=\sum_{k=1}^{∞}ξ_ka_k$, where $\sum_{k=1}^{∞}a_k$ is a convergent positive series and $ξ_k$ 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 $\sum_{k=1}^{∞}a_k$, 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 $ξ=\sum_{k=1}^{∞}ε_ka_k$, with $ε_k\in\{0, 1\}$. It is proved that $μ_ξ$ has either a pure discrete distribution or a pure singulary continuous distribution.