On the exact Hausdorff dimension of the set of Liouville numbers
type: Item
instance of: scholarly article
title: On the exact Hausdorff dimension of the set of Liouville numbers
zbMATH Open document ID: 1095.28012
author: L. Olsen
published in: Manuscripta Mathematica
publication date: 2005-04-14T00:00:00Z
review text: The set \(L\) of Liouville numbers -- composed of any irrational \(x\) such that, for all \(n\), there are integers \(p\) and \(q>1\) satisfying \(| x-{p\over q}| <{1\over q^n}\) -- is known to have Hausdorff dimension zero [\textit{J. C. Oxtoby}, ``Measure and category'' (1980; Zbl 0435.28011)]. A translation invariance argument shows that, for any dimension function \(h\), the \(h\)-dimensional Hausdorff measure is either zero or infinity. The main result in the paper amounts to partition the set of concave dimension functions in two classes, namely, if \(h\) approaches zero faster than some power function (specifically \(\limsup_{r\rightarrow 0+}{h(r)\over r^t}=0\) for some \(t>0\)), the \(h\)-dimensional Hausdorff measure of \(L\) is -- of course -- zero; if, on the contrary, \(h\) is a concave dimension function that approaches zero slower than any power function (\(\limsup_{r\rightarrow 0+}{h(r)\over r^t}>0\) for any \(t>0\)), then the \(h\)-dimensional Hausdorff measure of \(L\) is infinity.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 11K55
zbMATH DE Number: 2156089
zbMATH Keywords: Liouville numbers
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: Hausdorff measure
MaRDI profile type: Publication
full work available at URL: s00229-004-0528-z
OpenAlex ID: W4245662002
cites work: Q4375489
cites work: Q3996488
cites work: ON THE DIMENSIONLESSNESS OF INVARIANT SETS
cites work: Q3874454
cites work: Q5603915
DOI: 10.1007/S00229-004-0528-Z
description: scientific article; zbMATH DE number 2156089