On characterized subgroups of \(\mathbb{R}\) and \(\mathbb{R}/\mathbb{Z}\)
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 \(\sigma(\mathbb{R},\tau_{\textbf{u}}(\mathbb{R}))_{|\mathbb{Z}}=\sigma(\mathbb{Z},\mathfrak{t}_{\textbf{u}}(\mathbb{R}/\mathbb{Z})\) and \(\sigma(\mathbb{R},c_{\textbf{u}}(\mathbb{R}))_{|\mathbb{Z}}=\sigma(\mathbb{Z},c_{\textbf{u}}(\mathbb{R}/\mathbb{Z})\). Finally, if \(\textbf{u}\) is a subsequence of \(\textbf{w}\) in \(\mathbb{R}_+\), some conditions that implies that \(\tau_{\textbf{u}}(\mathbb{R})\nsubseteq c_{\textbf{w}}(\mathbb{R})\) are shown, and then the index \((\tau_{\textbf{u}}(\mathbb{R}):\tau_{\textbf{u}}(\mathbb{R})\cap c_{\textbf{w}}(\mathbb{R}))>\aleph_0\).
scientific article; zbMATH DE number 7438458
scientific article