Infinite Bernoulli convolutions generated by multigeometric series and their properties

Abstract

The paper is devoted to infinite Bernoulli convolutions generated by positive multigeometric series and to probability distributions of random variables whose digits in an even integer base-$$$s$$$ expansion with two redundant digits form a sequence of independent and identically distributed random variables.
The main objects of the article are random variables: $$$\xi=\sum\limits_{n=1}^{\infty}\frac{\xi_n}{s^n}$$$, where $$$(\xi_n)$$$ is a sequence of independent and identically distributed random variables taking values $$$0, 1, 2, \dots, s-1, s, s+1$$$ with probabilities $$$p_0$$$, $$$p_1$$$, $$$p_2, \dots, p_{s-1}, p_s, p_{s+1}$$$ respectively $$$(3<s \in \mathbb{N})$$$; $$\eta=\sum\limits_{n=1}^{\infty}\left[\frac{3\eta_{(n-1)(m+1)+1}}{s^n}+\sum\limits_{j=1}^{m} \frac{2\eta_{(n-1)(m+1)+1+j}}{s^n}\right]$$ where $$$(\eta_n)$$$ is a sequence of independent and identically distributed random variables that take values 0 and 1 with probabilities $$$q_0>0$$$ and $$$q_1=1-q_0>0$$$. We study conditions under which the above random variables have absolutely continuous or singular distributions as well as topological, metric, and fractal properties of their supports. The main focus is on the case where the spectrum is a Cantorval.
For the case $$$s=4$$$, we establish necessary and sufficient conditions for singularity and absolute continuity of the distributions of $$$\xi$$$ and $$$\eta$$$, in particular when they are supported on the Guthrie–Nymann Cantorval. For an arbitrary even $$$s>4$$$, we determine necessary and sufficient conditions for a random variable $$$\xi$$$ to admit a decomposition into the sum of two independent random variables, one of which is uniformly distributed on the unit interval and therefore absolutely continuous. Using the method of characteristic functions, sufficient conditions for singularity and necessary conditions for absolute continuity are obtained. For Cantorvals arising as spectra of the corresponding distributions, the structure and fractal properties of the boundary are investigated.