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

TL;DR

This paper explores the structure of infinite torsion subgroups of the circle, showing they can be characterized by bounded-ratio arithmetic-type sequences and revealing limitations of existing dichotomies.

Contribution

It extends previous work by characterizing infinite torsion subgroups via arithmetic-type sequences and analyzing the applicability of Eggleston's theorem to these sequences.

Findings

Infinite torsion subgroups are characterized by bounded-ratio arithmetic-type sequences.

Countability of characterized subgroups corresponds to torsion property.

Eggleston's dichotomy does not generally apply to arithmetic-type sequences.

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.