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

TL;DR

This paper investigates the Hausdorff dimension of exceptional sets in $ heta$-expansions, extending classical results on badly approximable numbers and partial quotients, and characterizing the size of sets with specific growth rates of partial quotients.

Contribution

It extends Jarnik's results to $ heta$-expansions and proves that sets with prescribed growth rates of partial quotients have full Hausdorff dimension.

Findings

Sets of numbers with specific partial quotient growth rates have full Hausdorff dimension.

Generalization of classical Diophantine approximation results to $ heta$-expansions.

Characterization of the Hausdorff dimension of exceptional sets in $ heta$-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.