# Representations of real numbers induced by probability distributions on $\mathbb{N}$

## TL;DR

This paper explores how probability distributions on natural numbers induce unique representations of real numbers in [0,1), analyzing their Hausdorff dimensions and digit frequency properties, with examples including geometric, Poisson, and zeta distributions.

## Contribution

It introduces a novel framework linking probability distributions on nd numbers to real number representations and studies their fractal and frequency characteristics.

## Key findings

- Hausdorff dimension of digit-restricted sets determined
- Prevalent digit frequencies identified for these representations
- Explicit examples with geometric, Poisson, and zeta distributions analyzed

## Abstract

We observe that a probability distribution supported by $\mathbb{N}$, induces a representation of real numbers in [0, 1) with digits in $\mathbb{N}$. We first study the Hausdorff dimension of sets with prescribed digits with respect to these representations. Than we determine the prevalent frequency of digits and the Hausdorff dimension of sets with prescribed frequencies of digits. As examples we consider the geometric distribution, the Poisson distribution and the zeta distribution.