A quest for countable statistically characterized subgroups.

‎‎‎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 subgroups coincide with the corresponding characterized subgroups. 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. Finally, we show that these results cannot be extended to general arithmetic-type sequences. .