Hausdorff dimension of exceptional sets arising in $\theta$-expansions

For a fixed ${\theta }^2=1\mathrm{/}m$, $m\in {\mathbb{N}}_{+}$, let $x\in [0,\theta )$ and $[a_1(x)\theta ,a_2(x)\theta ,\dots ]$ be the $\theta$-expansion of $x$. Our first goal is to extend for $\theta$-expansions the results of Jarnik \cite{J-1928} concerning the set of badly aproximable numbers and the set of irrationals whose partial quotients do not exceed a positive integer. Define $L_n(x)={\displaystyle \underset{1\le i\le n}{\max }{a}_i(x),x\in \mathrm{\Omega }:=[0,\theta )\setminus \mathbb{Q}}$. The second goal is to complete our result inspired by Philipp \cite{Ph-1976} % $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{L_n(x)\log \log n}{n}=\frac{1}{\log (1+{\theta }^2)}fora\mathrm{.}e\mathrm{.}x\in [0,\theta ]\mathrm{.}$$ % In this regard we prove that for any $\eta >0$ the set $$E(\eta )=\{x\in \mathrm{\Omega }:\underset{n\to \mathrm{\infty }}{\lim }\frac{L_n(x)\log \log n}{n}=\eta \}$$ is of full Hausdorff dimension.