On the exact Hausdorff dimension of the set of Liouville numbers. II
type: Item
instance of: scholarly article
title: On the exact Hausdorff dimension of the set of Liouville numbers. II
zbMATH Open document ID: 1126.28007
author: L. Olsen
author: Dave L. Renfro
published in: Manuscripta Mathematica
publication date: 2006-03-21T00:00:00Z
review text: This paper extends a previous resultby the first author [cf. Part I, Manuscr. Math. 116, No.~2, 157--172 (2005; Zbl 1095.28012)] on the classification of dimension functions for which the associated \(h\)-Hausdorff measure of the set of Liouville numbers \(L\) is either null or infinite. The authors prove here that for any arbitrary dimension function \(h\), either \(L\) has zero \(h\)-Hausdorff measure or \(L\) cannot be decomposed in countably many sets of finite \(h\)-Hausdorff measure (in particular, \(L\) has infinite \(h\)-Hausdorff measure). The dichotomy depends on the potential growth of the function \(\Gamma(r):=\inf_{0<s\leq r}rh(s)/s\) as \(r\) goes to 0.
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 5013735
zbMATH Keywords: Liouville numbers
zbMATH Keywords: Hausdorff dimension
Wikidata QID: Q56091111
MaRDI profile type: Publication
full work available at URL: s00229-005-0604-z
OpenAlex ID: W2135486963
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cites work: ON THE DIMENSIONLESSNESS OF INVARIANT SETS
cites work: On the exact Hausdorff dimension of the set of Liouville numbers
cites work: Q3874454
DOI: 10.1007/S00229-005-0604-Z
description: scientific article; zbMATH DE number 5013735