Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences
Number Theory
2025-06-19 v1 Group Theory
Abstract
In a 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 the circle 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, Eggleston, 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.
Cite
@article{arxiv.2506.15257,
title = {Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences},
author = {Ayan Ghosh and Pratulananda Das},
journal= {arXiv preprint arXiv:2506.15257},
year = {2025}
}