Hausdorff measures of subgroups of \(\mathbb{R}/\mathbb{Z}\) and \(\mathbb{R}\)
type: Item
instance of: scholarly article
title: Hausdorff measures of subgroups of \(\mathbb{R}/\mathbb{Z}\) and \(\mathbb{R}\)
zbMATH Open document ID: 1305.28013
author: Hans Weber
author: Enrico Zoli
published in: Ricerche di Matematica
publication date: 2014-12-08T00:00:00Z
review text: Let \(u=(u_n)_{n\in{\mathbb N}}\) be a sequence in \(]0, +\infty[\). The authors study the Hausdorff measure of the group \[ \tau_u =\{x\in{\mathbb R}: u_{n}x\to 0 \quad \text{modulo} \quad 1\}. \] Let \({\mathcal H}^f\) stand for the Hausdorff measure generated by an increasing function \(f\). The main result of the paper is the construction of the sequence \(u\) such that \({\mathcal H}^{f}(\tau_u)=+\infty\) and \({\mathcal H}^{g}(\tau_u)=0\) for a given pair of functions \(f\), \(g\) such that \(f(t)/g(t) \to +\infty\), \(f(t)/t \to +\infty\) and \(g(t)/t \to +\infty\) for \(t\to +0\).
reviewed by: Boris A. Kats
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 22B99
zbMATH DE Number: 6378221
zbMATH Keywords: Hausdorff measures
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: subgroups of the reals
zbMATH Keywords: subgroups of the circle group
zbMATH Keywords: topological torsion
MaRDI profile type: Publication
full work available at URL: s11587-013-0156-6
OpenAlex ID: W1984572873
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DOI: 10.1007/S11587-013-0156-6
description: scientific article; zbMATH DE number 6378221