The dimension of irregular set in parameter space

TL;DR

This paper investigates the Hausdorff dimension of sets in the parameter space of real numbers greater than one, characterized by the growth rate of the length of certain cylinders related to $eta$-expansions of 1.

Contribution

It determines the Hausdorff dimension of the set of parameters with a specified growth rate of cylinder lengths in $eta$-expansions, advancing understanding of the fractal structure of parameter spaces.

Findings

Hausdorff dimension of sets with given growth rates is explicitly calculated

Growth rates of cylinder lengths are characterized in the parameter space

Provides a detailed fractal analysis of $eta$-expansion parameter sets

Abstract

For any real number $\beta>1$. The $n$th cylinder of $\beta$ in the parameter space $\{\beta\in \mathbb{R}: \beta>1\}$ is a set of real numbers in $(1,\infty)$ having the same first $n$ digits in their $\beta$-expansion of $1$, denote by $I^P_n(\beta)$. We study the quantities which describe the growth of the length of $I^P_n(\beta)$. The Huasdorff dimension of the set of given growth rate of the length of $I^P_n(\beta)$ will be determined in this paper.