A remark on continued fractions with sequences of partial quotients
type: Item
instance of: scholarly article
title: A remark on continued fractions with sequences of partial quotients
zbMATH Open document ID: 1192.11048
author: Jun Wu
published in: Journal of Number Theory
publication date: 2009-06-17T00:00:00Z
review text: The author studies the Hausdorff dimension of the set of real numbers \(x\) such that the regular continued fraction expansion of \(x\) has the properties that (i) all partial quotients lie in some subset \(B \subseteq {\mathbb N}\) and (ii) the \(n\)th partial quotient \(a_n(x)\) satisfies the inequality \(a_n(x) \geq f(n)\), where \(f: {\mathbb N} \rightarrow {\mathbb R}\) is some function which tends to infinity. It was shown by \textit{B.-W. Wang} and \textit{J. Wu} [Bull. Lond. Math. Soc. 40, No. 1, 18--22 (2008; Zbl 1155.11046)] that without condition (ii), the Hausdorff dimension of this set is \(\tau(B)/2\), where \(\tau(B)\) is defined as \(\inf\{s > 0 : \sum_{i=1}^\infty b_i^{-s} < \infty\}\) for \(B = \{b_i\}_{i=1}^\infty\). In this paper, given a function \(f\) tending to infinity, the author constructs a set \(B\) with \(\tau(B) = 1\), but such that the Hausdorff dimension of the corresponding set of real numbers is equal to zero. This shows that the result of \textit{Wang} and \textit{Wu} [loc.cit.] is sharp.
reviewed by: Simon Kristensen
Mathematics Subject Classification ID: 11K50
zbMATH DE Number: 5565655
zbMATH Keywords: Continued fractions
zbMATH Keywords: restricted partial quotients
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.12.004
OpenAlex ID: W2059990188
cites work: HAUSDORFF DIMENSION OF SETS OF CONTINUED FRACTIONS
cites work: Q5838540
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DOI: 10.1016/J.JNT.2007.12.004
description: scientific article; zbMATH DE number 5565655