Hausdorff dimension for the set of points connected with the generalized Jarn\'ik-Besicovitch set

TL;DR

This paper investigates the Hausdorff dimension of a generalized set related to continued fractions, extending classical results and recent research by analyzing the growth of partial quotients under certain conditions.

Contribution

It introduces a generalized framework for the Hausdorff dimension of sets defined by continued fraction partial quotients, encompassing classical and recent theorems as special cases.

Findings

Derived formulas for Hausdorff dimension under various conditions

Unified classical and recent results within a generalized setting

Extended understanding of the structure of sets defined by continued fractions

Abstract

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in\mathbb{N},$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} {align*} holds for infinitely many $n\in\mathbb{N},$ where $h$ and $\tau$ are positive continuous functions, $T$ is the Gauss map and $a_n(x)$ denote the $n$th partial quotient of $x$ in its continued fraction expansion. By appropriate choices of $r$, $\tau(x)$ snd $h(x)$ we obtain the classical Jarn\'{i}k-Besicovitch Theorem as well as more recent results by Wang-Wu-Xu, Wang-Wu, Huang-Wu-Xu and Hussain-Kleinbock-Wadleigh-Wang.