Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909652376092672 |
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| author | Ghosh, Ayan Das, Pratulananda |
| author_facet | Ghosh, Ayan Das, Pratulananda |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_15257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences Ghosh, Ayan Das, Pratulananda Number Theory Group Theory 11J71 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. |
| title | Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences |
| topic | Number Theory Group Theory 11J71 |
| url | https://arxiv.org/abs/2506.15257 |