Singular distributions of random variables with independent digits of representation in numeral system with natural base and redundant alphabet

Given natural parameters $s$ and $r$, where $2\leq s\leq r$, we consider the distribution of a~ran\-dom variable $$\xi=\sum\limits_{k=1}^{\infty}s^{-k}\xi_k\equiv\Delta^{r_s}_{\xi_1\xi_2...\xi_k...},$$where $(\xi_k)$ is a sequence of independent random variables taking values in $\{0,1,...,r\}$ with probabilities $p_0,$ $p_1,$ $\ldots,$ $p_r$, respectively, and all $ p_i<1$. In the case $s=3=r$, necessary and sufficient conditions for the singularity and absolute continuity of the distribution of $\xi$ are established. It is shown that the distribution of $\xi$ is absolutely continuous if and only if $p_1=\frac{1}{3}=p_2$. In all other cases, the distribution is singular (i.e., supported on a set of zero Lebesgue measure).For $p_0p_1p_2p_3=0$, the fractal Hausdorff–Besicovitch dimension of the spectrum (i.e., the minimal closed support) of the distribution of $\xi$ and of the essential support of its density is explicitly determined under the condition $p_{i}p_{i+1}p_{i+2}\ne0$ for $i=0,1$. The work also discusses the connection between the distribution of $\xi$ and infinite Bernoulli convolutions governed by the corresponding series as well as representations of numbers in the base-$3$ numeral system with one redundant digit. Several open problems are formulated. For the numeral system with the base $3$ and the alphabet $A=\{0,1,2,3\}$, the problem of determining the number of representations of a number is completely solved. It is proven that almost all numbers (with respect to the Lebesgue measure) in the interval $[0;\frac{3}{2}]$ have a~continuum of distinct representations, while those with a unique representation form a fractal set of Hausdorff–Besicovitch dimension $\log_3{2}$.