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

Pratsiovytyi Mykola; Makarchuk Oleg; Karvatskyi Dmytro

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arXiv · 2412.17430

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

Pratsiovytyi Mykola·Makarchuk Oleg·Karvatskyi Dmytro

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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.

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BibTeX RIS

Pratsiovytyi Mykola, Makarchuk Oleg, Karvatskyi Dmytro. 2024-12-23. On the Lebesgue structure of the distribution of a random variable defined by continued $A_2$-fractions. https://arxiv.org/abs/2412.17430

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