Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences

Ayan Ghosh; Pratulananda Das

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arXiv · 2506.15257

Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences

Ayan Ghosh·Pratulananda Das

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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.

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Ayan Ghosh, Pratulananda Das. 2025-06-18. Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences. https://arxiv.org/abs/2506.15257

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