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HAUSDORFF DIMENSION FOR THE SET OF POINTS CONNECTED WITH THE GENERALIZED JARNÍK–BESICOVITCH SET

Part of: Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms Classical measure theory

Published online by Cambridge University Press:  07 December 2020

AYREENA BAKHTAWAR
AYREENA BAKHTAWAR*
Affiliation:
Department of Mathematics and Statistics, La Trobe University, PO Box 199, Bendigo, Victoria 3552, Australia e-mail: a.bakhtawar@latrobe.edu.au
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Abstract

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$,

$$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$
holds for infinitely many $n\in \mathbb {N}$, where h and $\tau $ are positive continuous functions, T is the Gauss map and $a_{n}(x)$ denotes the nth partial quotient of x in its continued fraction expansion. By appropriate choices of $r,\tau (x)$ and $h(x)$ we obtain various classical results including the famous Jarník–Besicovitch theorem.

Keywords

continued fractionsHausdorff dimensionJarník–Besicovitch theorem

MSC classification

Primary: 11K50: Metric theory of continued fractions
Secondary: 11J70: Continued fractions and generalizations 11J83: Metric theory 28A78: Hausdorff and packing measures

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 1 , February 2022 , pp. 1 - 29
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Dzmitry Badziahin

This research was supported by a La Trobe University Postgraduate Research Award.

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