$\widetilde{Q}$-representation of real numbers and fractal probability distributions

TL;DR

This paper introduces a new $ ilde{Q}$-representation for real numbers, extending existing representations, and explores its applications in fractal geometry and the detailed probabilistic properties of associated distributions.

Contribution

It develops the $ ilde{Q}$-representation as a versatile tool for fractal analysis and characterizes the types of probability measures arising from random variables with independent $ ilde{Q}$-symbols.

Findings

Conditions for absolute continuity or singularity of measures are established.

The metric-topological properties of distributions are analyzed.

Examples illustrating the theoretical results are provided.

Abstract

A $\widetilde{Q}-$representation of real numbers is introduced as a generalization of the $p-$adic and $Q-$representations. It is shown that the $\widetilde{Q}-$representation may be used as a convenient tool for the construction and study of fractals and sets with complicated local structure. Distributions of random variables $\xi$ with independent $\widetilde{Q}-$symbols are studied in details. Necessary and sufficient conditions for the probability measures $\mu_\xi $ associated with $\xi$ to be either absolutely continuous or singular (resp. pure continuous, or pure point) are found in terms of the $\widetilde{Q}-$representation. In addition the metric-topological properties for the distribution of $\xi $ are investigated. A number of examples are presented.