Statistically Characterized Subgroups: A Comparison Between Arithmetic and Non-Arithmetic Sequences.

Very recently, characterized subgroups associated with certain non-arithmetic sequences were studied in [Das et al., Mediterr. J. Math. 21:164, 2024]. On the other hand, statistically characterized subgroups introduced as a non-trivial generalization of characterized subgroups have been primarily explored in the context of arithmetic sequences [Dikranjan et al., Fund. Math. 249:185 – 209, 2020]. This naturally leads to the question of how such subgroups behave when defined via non-arithmetic sequences, which is the focus of the present work.

Our investigation reveals that statistically characterized subgroups corresponding to these non-arithmetic sequences are mostly larger in size, typically of cardinality c, and exhibit properties markedly distinct from those of classical characterized subgroups. In particular, it was shown in [Das et al., Bull. Sci. Math., 103157, 2022] that for arithmetic sequences, statistically characterized subgroups are never characterizable. However, by employing certain non-arithmetic sequences, we construct for the first time, a countable (and hence characterizable) statistically characterized subgroup.

Some of the results presented in this talk have been published in Expositiones Mathematicae (2025)