Multifractal aspects of $\alpha$-expansions

TL;DR

This paper investigates the multifractal properties of alpha-expansions, a class of real number representations, extending previous work on their ergodic and dimension theoretical aspects.

Contribution

It provides a detailed analysis of the multifractal structure of alpha-expansions, revealing new insights into their complex geometric and measure-theoretic behavior.

Findings

Identifies multifractal spectra associated with alpha-expansions

Establishes connections between multifractal analysis and ergodic properties

Provides new dimension estimates for sets defined by alpha-expansions

Abstract

In \cite{[NE]} we introduce $\alpha$-expansions a real numbers in $(0,1]$, given by \[ \sum_{i=1}^{\infty}(\alpha-1)^{i-1}\alpha^{-(d_{1}+\dots+d_{i})}\] with $\alpha>1$ and $d_{i}\in\mathbb{N}$ and discuss ergodic theoretical and dimension theoretical aspects of this expansions. In this sequel we study mutifractal aspects of this expansions.