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

TL;DR

This paper investigates the conditions under which the distribution of a randomly constructed number in a redundant base system is singular or absolutely continuous, linking it to Bernoulli convolutions and numeral representations.

Contribution

It establishes necessary and sufficient conditions for singularity and absolute continuity of distributions in redundant numeral systems with independent digits.

Findings

Identifies conditions for singularity and absolute continuity in specific base systems.

Connects the distribution properties to Bernoulli convolutions.

Formulates several open problems in the area.

Abstract

Given natural parameters s and r, where $2\leq s\leq r$, we consider the distribution of a random 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 $p_0,p_1,...,p_r$, 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 random variable are established. The work also discusses the connection between the distribution of random variable 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.