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

  • Hans WeberEmail author


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 \({\mathbb R}\).


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

Mathematics Subject Classification

22A10 43A40 11K06 


Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interest.


  1. 1.
    Barbieri, G., Dikranjan, D., Giordano Bruno, A., Weber, H.: Dirichlet sets vs characterized subgroups. Topol. Appl. 231, 50–76 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barbieri, G., Dikranjan, D., Milan, C., Weber, H.: Answer to Raczkowki’s questions on convergent sequences of integers. Topol. Appl. 132, 89–101 (2003)CrossRefGoogle Scholar
  3. 3.
    Barbieri, G., Giordano Bruno, A., Weber, H.: Inclusions of characterized subgroups. Topol. Appl. 221, 534–555 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borel, J.-P.: Sous-groupes de \(\mathbb{R}\) liés à la répartition modulo 1 de suites. Ann. Fac. Sci. Toulouse Math. 5, 217–235 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Comfort, W.W., Ross, K.A.: Topologies induced by groups of characters. Fund. Math. 55, 283–291 (1964)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dikranjan D., Das P., Bose K.: Statistically characterized subgroups of the circle. Fund. Math. (to appear) Google Scholar
  7. 7.
    Di Santo, R., Dikranjan, D., Giordano Bruno, A.: Characterized subgroups of the circle group. Ric. Mat. 67, 625–655 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Eggleston, H.G.: Sets of fractional dimensions which occur in some problems of number theory. Proc. Lond. Math. Soc. 54, 42–93 (1952)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Elias, P.: Arbault permitted sets are perfectly meager. Tatra Mt. Math. Publ. 30, 135–148 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Flor, P.: Zur Bohr-Konvergenz von Folgen. Math. Scand. 23, 169–170 (1968)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley, New York (1974)zbMATHGoogle Scholar
  12. 12.
    Weber, H., Zoli, E.: Hausdorff measures of subgroups of \(\mathbb{R}/\mathbb{Z}\) and \(\mathbb{R}\). Ric. Mat. 62, 209–228 (2013)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations