Characterizing infinite torsion subgroups of the circle via arithmetic‐type sequences

A subgroup of the circle group is called characterized by a sequence of integers if . Borel ( Ann. Fac. Sci. Toulouse Math . 5 (1983), 217–235) was able to establish that every countable subgroup of can be characterized but his result remains purely existential even for the best known class of countable subgroups, namely, the class of torsion subgroups of . With this question in mind, a new class of sequences called “arithmetic‐type sequences” was introduced and in a very recent work (Das et al., Bull. Sci. Math . 199 (2025), 103580), the structure of characterized subgroups corresponding to arithmetic‐type sequences was investigated. Building upon this work, we further show that a characterized subgroup associated with an arithmetic‐type sequence is countable if and only if it is torsion. Further, we prove that any infinite torsion subgroup of can be characterized by an arithmetic‐type sequence with bounded ratio. Moreover, our findings demonstrate that the dichotomy observed in Eggleston's theorem (Theorem 16, Proc. Lond. Math. Soc . 54(2) (1952), 42–93) for arithmetic sequences does not extend, in general, to the broader class of arithmetic‐type sequences.