The dimension of irregular set in parameter space

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.