Infinite Bernoulli convolutions generated by multigeometric series and their properties

TL;DR

This paper investigates the properties of infinite Bernoulli convolutions generated by multigeometric series, analyzing conditions for absolute continuity or singularity and exploring the fractal structure of their supports, including Cantorvals.

Contribution

It introduces new conditions for the distribution types of these convolutions and examines their topological and fractal properties, especially focusing on Cantorval spectra.

Findings

Identifies conditions for absolute continuity and singularity of distributions.

Analyzes the topological and fractal structure of the support sets.

Characterizes the spectrum as a Cantorval under certain conditions.

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.