Statistically characterized subgroups of the circle
Volume 249 / 2020
Fundamenta Mathematicae 249 (2020), 185-209
MSC: Primary 22B05; Secondary 11B05, 40A05.
DOI: 10.4064/fm690-8-2019
Published online: 13 December 2019
Abstract
In this paper, we introduce a new version of characterized subgroups of the circle group $\mathbb T $ that we call statistically characterized subgroups (briefly, s-characterized subgroups). We primarily investigate these subgroups for arithmetic sequences of integers and show that these subgroups are essentially different and strictly larger in size than the much investigated class of characterized subgroups, having cardinality $\mathfrak {c}$ but remaining nontrivial (i.e. different from $\mathbb T $) though remaining topologically nice.
