The growth rate and dimension theory of beta-expansions

TL;DR

This paper investigates the growth rate and fractal dimension properties of beta-expansions, extending previous results on exponential growth and analyzing their dimensional characteristics.

Contribution

It advances the understanding of beta-expansions by analyzing their growth rates and dimension theory beyond previous exponential growth results.

Findings

Exponential growth of beta-expansions for certain beta values.

Dimension properties of beta-expansions are characterized.

Extended the theoretical framework of beta-expansion analysis.

Abstract

In a recent paper of Feng and Sidorov they show that for $\beta\in(1,\frac{1+\sqrt{5}}{2})$ the set of $\beta$-expansions grows exponentially for every $x\in(0,\frac{1}{\beta-1})$. In this paper we study this growth rate further. We also consider the set of $\beta$-expansions from a dimension theory perspective.