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On characterized subgroups of \(\pmb {{\mathbb {R}}}\) and \(\pmb {{\mathbb {R}}}/\pmb {{\mathbb {Z}}}\)

  • Hans WeberEmail author
Article

Abstract

For a real-valued sequence \(\mathbf u \) let \(\tau _\mathbf{u }({\mathbb R})=\{x\in {\mathbb R}: u_nx\rightarrow 0 \text { mod }{\mathbb Z}\}\). In Barbieri et al. (Topol Appl 221:534–555, 2017) there were given conditions which imply that \(\tau _\mathbf{u }({\mathbb R})\nsubseteq \tau _\mathbf{v }({\mathbb R})\). Here we positively answer a question posed in Barbieri et al. (2017) and then in Di Santo et al. (Ric Mat 67:625–655, 2018) showing that these conditions imply that even \((\tau _\mathbf{u }({\mathbb R}):\tau _\mathbf{u }({\mathbb R})\cap \tau _\mathbf{v }({\mathbb R}))>\aleph _0\). Hereby the group \(c_\mathbf{u }({\mathbb R}):=\{x\in {\mathbb R}: (u_nx)_{n\in {\mathbb N}} \text { converges mod }{\mathbb Z}\text { in }{\mathbb R}\}\) plays an important role. We also present some basic properties of \(c_\mathbf{u }({\mathbb R})\), the relationship between \(c_\mathbf{u }({\mathbb R})\) and \(\tau _\mathbf{u }({\mathbb R})\) and the connection between these groups and certain totally bounded group topologies on \({\mathbb Z}\) and \({\mathbb R}\).

Keywords

Characterized subgroups Convergence modulo 1 of real sequences Totally bounded group topologies Weak inclusion 

Mathematics Subject Classification

22A10 43A40 11K06 

Notes

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interest.

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Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Dipartimento di Scienze Matematiche, Informatiche e FisicheUniversità degli Studi di UdineUdineItaly

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