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

Jörg Neunhäuserer

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.

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

Abstract

We observe that a probability distribution supported by , induces a representation of real numbers in [0, 1) with digits in . 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.

Paper Structure

This paper contains 3 sections, 6 theorems, 39 equations.

Table of Contents

  1. The Representations
  2. Prescribed digits
  3. Frequency of digits

Key Result

Proposition 1.1

For all probability distributions ${\mathfrak p}$ supported by $\mathbb{N}$, the map $\pi_{\mathfrak{p}}:\mathbb{N}^\mathbb{N}\to [0,1)$ is a bijection. Moreover the map $\pi_{\mathfrak{p}}$ is continuous, if we endorse $\mathbb{N}^\mathbb{N}$ with the metric where $\delta(x,y)=0$ if $x=y$ and $\delta(x,y)=1$ otherwise.

Theorems & Definitions (6)

  • Proposition 1.1
  • Theorem \oldthetheorem
  • Corollary 2.1
  • Theorem \oldthetheorem
  • Corollary 3.1
  • Theorem \oldthetheorem