# On singularity of distribution of random variables with independent symbols of Oppenheim expansions

## TL;DR

This paper investigates the distribution properties of real numbers' expansions in the restricted Oppenheim framework, establishing conditions for singularity and finiteness of digit appearances in almost all cases.

## Contribution

It introduces new conditions for the singularity of distributions of random variables with independent symbols in restricted Oppenheim expansions, extending to non-i.i.d. cases.

## Key findings

- Almost all real numbers have expansions with finitely many occurrences of any digit.
- Distributions of i.i.d. symbol-based random variables are singular relative to Lebesgue measure.
- Sufficient conditions for singularity are provided for non-i.i.d. cases.

## Abstract

The paper is devoted to restricted Oppenheim expansion of real numbers ($ROE$),which includes as partial cases already known Engel, Silvester and L\"uroth expansions. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their $ROE$-expansion contain arbitrary digit $i$ only finitely many times. Main results of the paper states the singularity (w.r.t. Lebesgue measure) of the distribution of random variable with i.i.d increments of symbols of restricted Oppenheim expansion. General non-i.i.d. case are also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.