Singularity and fine fractal properties of a certain class of infinite Bernoulli convolutions with an essential intersection

We prove that the probability distribution of a random variable $\xi$ represented in the form of an infinite series \[ \xi = \sum _{k=1}^\infty \xi _k a_k \] is singular, where $\xi _k$ are independent Bernoulli random variables and where the sequence $\{a_k\}$ is such that the series $\sum _{k=1}^\infty a_k$ converges, $a_k\ge 0$ for all $k\ge 1$, and, for an arbitrary $k\in \mathbf {N}$, there exists $s_{k} \in \mathbf {N}\cup \{0\}$ for which $s_k>0$ for infinitely many indices $k$ and $a_{k} = a_{k+1} = \dots = a_{k+s_{k} } \ge r_{k+s_{k} }$, where $r_k$ is the tail of the series, namely \[ r_k=\sum _{i=k+1}^\infty a_i. \] Under these assumptions, it is shown that the corresponding distribution is a Bernoulli convolution with essential intersections (that is, almost all with respect to the Hausdorff–Besicovitch dimension points of the spectrum have continuum many different expansions of the form $\sum _{k=1}^\infty \omega _k a_k$, where $\omega _k \in \{0,1\}$). Our main attention is paid to the studies of fractal properties of singularly continuous probability measures $\mu _{\xi }$. In particular, fractal properties of the spectra (minimal closed supports of the above measures) and minimal in the sense of the Hausdorff–Besicovitch dimension dimensional supports of such probability distributions are studied in detail.