The Hausdorff Dimension of the Sets of Irrationals with Prescribed Relative Growth Rates

For a fixed \(\vartheta ^2=1/m\) ${\vartheta }^2=1/m$ , \(m \in {\mathbb {N}}_+\) $m\in {\mathbb{N}}_{+}$ , let \(x \in [0, \vartheta )\) $x\in [0,\vartheta )$ and \([b_1(x) \vartheta , b_2(x) \vartheta , \ldots ]\) $[b_1(x)\vartheta ,b_2(x)\vartheta ,\dots ]$ be the \(\vartheta \) $\vartheta$ -expansion of x. In this paper, our objective is to investigate the relationship between the growth rate of the largest digit and the convergence rate for the \(\vartheta \) $\vartheta$ -expansion of an irrational number \(x \in \Omega := [0, \vartheta ) \setminus {\mathbb {Q}}\) $x\in \mathrm{\Omega }:=[0,\vartheta )\backslash \mathbb{Q}$ . We demonstrate that the Hausdorff dimension of the set comprising irrationals with a specified relative growth rate is complete.