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

Abstract

For a fixed $\theta^2=1/m$, $m \in \mathbb{N}_+$, let $x \in [0, \theta)$ and $[a_1(x) \theta, a_2(x) \theta, \ldots]$ 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 \max_{1 \leq i \leq n} a_i(x), x \in \Omega:=[0, \theta)\setminus \mathbb{Q} $. The second goal is to complete our result inspired by Philipp \cite{Ph-1976} % \[ \liminf_{n \to \infty} \frac{L_n(x) \log\log n}{n} = \frac{1}{\log \left( 1+ \theta^2\right)} \mbox{ for a.e. } x \in [0, \theta]. \] % In this regard we prove that for any $\eta > 0$ the set \[ E(\eta) = \left\lbrace x \in \Omega: \lim_{n \to \infty} \frac{L_n(x) \log\log n}{n} = \eta \right\rbrace \] is of full Hausdorff dimension.