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 $[a_1(x)\theta ,a_2(x)\theta ,\dots ]$ 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 $L_n(x)=\underset{1\le i\le n}{\max }{a}_i(x),x\in \Omega :=[0,\theta )\setminus \mathbb{Q}$. The second goal is to complete our result inspired by Philipp \cite{Ph-1976} % \[ \liminf_{n \to \infty} \frac{L_n(x) \log\log n}{n} = \frac{1}{\log \left( 1+ θ^2\right)} \mbox{ for a.e. } x \in [0, θ]. \] % In this regard we prove that for any $\eta >0$ the set \[ E(η) = \left\lbrace x \in Ω: \lim_{n \to \infty} \frac{L_n(x) \log\log n}{n} = η\right\rbrace \] is of full Hausdorff dimension.

arXiv comment17 pages