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 $(1,\mathrm{\infty })$ having the same first $n$ digits in their $\beta$-expansion of $1$, denote by $I_n^P(\beta )$. We study the quantities which describe the growth of the length of $I_n^P(\beta )$. The Huasdorff dimension of the set of given growth rate of the length of $I_n^P(\beta )$ will be determined in this paper.