A new family of expansions of real numbers

TL;DR

This paper introduces a novel family of real number expansions parameterized by b1>1, exploring their ergodic, dimension theoretical properties, and base-change transformations, thus contributing new insights into number representation systems.

Contribution

It presents a new expansion method for real numbers in (0,1] and analyzes its ergodic, dimension, and transformation properties, which are previously unexplored.

Findings

Analysis of ergodic properties of the expansion.

Dimension theoretical characteristics studied.

Behavior of base-change transformations examined.

Abstract

For $\alpha>1$ we represent a real number in $(0,1]$ in the form \[ \sum_{i=1}^{\infty}(\alpha-1)^{i-1}\alpha^{-(d_{1}+\dots+d_{i})}\] with $d_{i}\in\mathbb{N}$. We discuss ergodic theoretical and dimension theoretical aspects of this expansion. Furthermore we study their base-change-transformation.