Statistically characterized subgroups related to some nonarithmetic sequence of integers II (characterization for countable subgroups)

Very recently in [Das et al., Expo. Math. 43(3):125653, 2025], statistically characterized subgroups have been investigated for certain types of nonarithmetic sequences. Building on this work, we investigate further and demonstrate that, for a particular class of nonarithmetic sequences, the statistically characterized subgroups coincide with the corresponding characterized subgroups. It had already been shown that statistically characterized subgroups corresponding to arithmetic sequences cannot be characterized by any sequence [Das et al., Bull. Sci. Math. 179(2):103157, 2022], and further, 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, that is, 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].