Statistically characterized subgroups related to some non-arithmetic sequence of integers II (a quest for countable subgroups)

TL;DR

This paper investigates the size and properties of statistically characterized subgroups related to non-arithmetic sequences, demonstrating that for certain sequences these subgroups can be countably infinite, thus answering open questions in the field.

Contribution

It shows that for a specific class of non-arithmetic sequences, statistically characterized subgroups are countably infinite, providing a negative answer to longstanding open problems.

Findings

Statistically characterized subgroups can be countably infinite.

For certain sequences, these subgroups coincide with characterized subgroups.

The results resolve several open problems in the literature.

Abstract

Following the work of [Dikranjan et al., Fund. Math. 249:185-209, 2020] for arithmetic sequences, very recently in [Das et al., Expo. Math. 43(3):125653, 2025], statistically characterized subgroups have been investigated for certain types of non-arithmetic sequences. Building on this work, we investigate further and demonstrate that, for a particular class of non-arithmetic sequences, the statistically characterized subgroup coincides with the corresponding characterized subgroup. In this context it should be kept in mind that statistical convergence (convergence w.r. to the ideal of natural density zero sets) encompasses much more sequences than usual convergence (convergence w.r. to the ideal of finite sets) and it had already been shown that statistically characterized subgroups corresponding to arithmetic sequences can not be characterized by any sequence [Das et al., Bull. Sci. Math. 179(2):103157, 2022] and they are always of the size of the continuum. From the very beginning it has been an open question as to whether statistically characterized subgroups can be small in size i.e. countably infinite. Our observation thus sheds new light on the crucial role of sequences generating subgroups of the circle group and at the same time one can subsequently identify a class of sequences for which statistically characterized subgroups are countably infinite. This result provides a negative solution to Problem 2.16 posed in [Das et al., Expo. Math. 43(3):125653, 2025] and Question 6.3 from [Dikranjan et al., Fund. Math. 249:185-209, 2020]. Additionally, our findings resolve several open problems from [Dikranjan et al., Topo. Appl., 2025].