Statistically characterized subgroups related to some non-arithmetic sequence of integers

TL;DR

This paper explores statistically characterized subgroups related to non-arithmetic integer sequences, revealing they are generally larger and behave differently from classically characterized subgroups, and solves an open problem in the field.

Contribution

It extends the study of statistically characterized subgroups to non-arithmetic sequences and addresses an open problem from prior research.

Findings

Statistically characterized subgroups are often of continuum cardinality.

These subgroups exhibit distinct behavior compared to classical characterized subgroups.

The paper solves an open problem related to these subgroups.

Abstract

Recently, in Das et al. (Mediterr. J. Math. 21 : 164, 2024), characterized subgroups are investigated for some special kind of non-arithmetic sequences. In this note, we study subsequent problems in case of ``statistically characterized subgroups" introduced in Dikranjan et al. (Fund. Math. 249 : 185-209, 2020). The entire investigation emphasizes that these statistically characterized subgroups are mostly larger in size, having cardinality $\mathfrak{c}$, and exhibit behavior that significantly differs from that of classically characterized subgroups. As a consequence, we solve an open problem raised in Dikranjan et al. (Fund. Math. 249 : 185-209, 2020).