Hausdorff Dimension of Growth Rate Level Sets in $\theta$-expansions

We investigate the Hausdorff dimension of level sets defined by digit growth rates in $\theta$-expansions, a generalization of regular continued fractions. For any $\alpha \ge 0$, we prove that the set $$E_{\theta }(\alpha )=\{x\in [0,\theta ]\setminus \mathbb{Q}:\underset{n\to +\mathrm{\infty }}{\lim }\frac{L_{n,\theta }(x)\log n\log \log n}{S_{n,\theta }(x)-L_{n,\theta }(x)}=\alpha \}$$ has full Hausdorff dimension. This extends previous work of Zhang and {L"u} (2016) on regular continued fractions to the broader framework of $\theta$-expansions. The proof involves constructing explicit subsets with controlled digit growth and establishing dimension preservation through H"older-continuous mappings.