On the Lebesgue structure of the distribution of a random variable defined by continued $A_2$-fractions

TL;DR

This paper investigates the Lebesgue structure of distributions derived from random variables expressed via $A_2$-continued fractions, establishing conditions for discreteness and singularity, and exploring metric properties.

Contribution

It provides necessary and sufficient conditions for the distribution's discreteness and sufficient conditions for singularity in the context of $A_2$-fractions.

Findings

Conditions for discreteness of the distribution

Conditions for singularity of the distribution

Analysis of metric properties of $A_2$-representations

Abstract

In this paper, we study the Lebesgue structure of the distribution of a random variable given in terms of a continued fraction with a two-symbol alphabet $\{\frac{1}{2}, 1\}$, also known as $A_2$-fractions. We establish necessary and sufficient conditions for the distribution to be discrete and provide some sufficient conditions for its singularity. We also explore non-trivial metric properties of the $A_2$-representation with the specified alphabet.