Hausdorff dimension for the set of points connected with the generalized Jarník-Besicovitch set

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 \(h\)...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Author Bakhtawar, Ayreena
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 16.10.2020
Subjects
Continuity (mathematics)
Quotients
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422