Characterized Subgroups Related to some Non-arithmetic Sequence of Integers

A subgroup H of the circle group \({\mathbb {T}}\) is called characterized by a sequence of integers \((u_n)\) if \(H=\{x\in {\mathbb {T}}: \lim _{n\rightarrow \infty } u_nx=0\}\). In this note, we primarily consider a non-arithmetic sequence arising out of an arithmetic sequence in line of Bíró et al. (Stud Sci Math Hung 38: 97–113, 2001) and investigate the corresponding characterized subgroup thoroughly which includes its cardinality aspects. The whole investigation reiterates that these characterized subgroups are infinitely generated unbounded torsion countable subgroups of the circle group \({\mathbb {T}}\). Finally, we delve into certain structure theoretic observations.