On characterized subgroups of \(\mathbb{R}\) and \(\mathbb{R}/\mathbb{Z}\)
type: Item
instance of: scholarly article
title: On characterized subgroups of \(\mathbb{R}\) and \(\mathbb{R}/\mathbb{Z}\)
zbMATH Open document ID: 1504.22003
author: Yanyan Li
published in: Ricerche di Matematica
publication date: 2021-12-03T00:00:00Z
review text: For \(\textbf{u}=(u_n)_{n\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}}\) let \(c_{\textbf{u}}(\mathbb{R})=\{x\in\mathbb{R}:\exists y\in\mathbb{R}\ (u_nx)_{n\in\mathbb{N}}\to y\mod \mathbb{Z}\}\) and \(\tau_{\textbf{u}}(\mathbb{R})=\{x\in\mathbb{R}: (u_nx)_{n\in\mathbb{N}}\to 0\mod \mathbb{Z}\}\). Groups of this type are called \emph{characterized subgroups} of \(\mathbb{R}\). Some basic properties of \(c_{\textbf{u}}(\mathbb{R})\) are presented. As example \(c_{\textbf{u}}(\mathbb{R})=\mathbb{R}\) if and only if \(\textbf{u}\) converges in \(\mathbb{R}\) (and \(\tau_{\textbf{u}}(\mathbb{R})=\mathbb{R}\) if and only if \(\textbf{u}\to 0\)), or \(c_{\textbf{u}}(\mathbb{R})\) is a Lebesgue null set if and only if \(\textbf{u}\) is not convergent. The relationship between \(c_{\textbf{u}}(\mathbb{R})\) and \(\tau_{\textbf{u}}(\mathbb{R})\) are also studied. The following possibilities can occur: the size of both, the index \((c_{\textbf{u}}(\mathbb{R}):\tau_{\textbf{u}}(\mathbb{R}))\) and \(\tau_{\textbf{u}}(\mathbb{R})\), can be small, both can be big or one of them can be small and the other one big. In this paper, the author presents the relationship between \(c_{\textbf{u}}(\mathbb{R})\) and certain totally bounded group topologies on \(\mathbb{Z}\) and on \(\mathbb{R}\). Then \(\tau_{\textbf{u}}(\mathbb{R})=c_{\textbf{u}}(\mathbb{R})\) if and only if \(u_n\to 0\, (\tau)\) provided \(\tau\) is a totally bounded group topology on \(\mathbb{R}\) weaker than \(\tau_{eu}\) and \(\textbf{u}\) is \(\tau\)-Cauchy. If \(\textbf{u}\) is a sequence in \(\mathbb{Z}\), the characerized subgroup \(\mathfrak{t}_{\textbf{u}}(\mathbb{R}/\mathbb{Z}):=\{t\in \mathbb{R}/\mathbb{Z}:u_nt\to 0\}=c_{\textbf{u}}(\mathbb{R})\) if and only if \(u_n\to 0\, (\tau)\) provided \(\tau\) is a totally bounded group topology on \(\mathbb{Z}\) such that \(\textbf{u}\) is \(\tau\)-Cauchy. In particular \(\sigma(\mathbb{R},\tau_{\textbf{u}}(\mathbb{R}))_{|\mathbb{Z}}=\sigma(\mathbb{Z},\mathfrak{t}_{\textbf{…
reviewed by: M. V. Ferrer
Mathematics Subject Classification ID: 22A10
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 7438458
zbMATH Keywords: characterized subgroups
zbMATH Keywords: convergence modulo 1 of real sequences
zbMATH Keywords: totally bounded group topologies
zbMATH Keywords: weak inclusion
MaRDI profile type: Publication
full work available at URL: s11587-019-00470-y
OpenAlex ID: W2987723578
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DOI: 10.1007/S11587-019-00470-Y
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description: scientific article; zbMATH DE number 7438458