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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.10826 |
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| _version_ | 1866917068298780672 |
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| author | Shor, Caleb M. Sim, Jae Hyung |
| author_facet | Shor, Caleb M. Sim, Jae Hyung |
| contents | In 2008, Wang \& Wang showed that the set of gaps of a numerical semigroup generated by two coprime positive integers $a$ and $b$ is equidistributed modulo 2 precisely when $a$ and $b$ are both odd. Shor generalized this in 2022, showing that the set of gaps of such a numerical semigroup is equidistributed modulo $m$ when $a$ and $b$ are coprime to $m$ and at least one of them is 1 modulo $m$. In this paper, we further generalize these results by considering numerical semigroups generalized by geometric sequences of the form $a^k, a^{k-1}b, \dots, b^k$, aiming to determine when the corresponding set of gaps is equidistributed modulo $m$. With elementary methods, we are able to obtain a result for $k=2$ and all $m$. We then work with cyclotomic rings, using results about multiplicative independence of cyclotomic units to obtain results for all $k$ and infinitely many $m$. Finally, we take an approach with cyclotomic units and Dirichlet L-functions to obtain results for all $k$ and all $m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10826 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equidistribution Conditions for Gaps of Geometric Numerical Semigroups Shor, Caleb M. Sim, Jae Hyung Number Theory 11D04, 20M14, 11F70, 11F66, 11R18, 11J82 In 2008, Wang \& Wang showed that the set of gaps of a numerical semigroup generated by two coprime positive integers $a$ and $b$ is equidistributed modulo 2 precisely when $a$ and $b$ are both odd. Shor generalized this in 2022, showing that the set of gaps of such a numerical semigroup is equidistributed modulo $m$ when $a$ and $b$ are coprime to $m$ and at least one of them is 1 modulo $m$. In this paper, we further generalize these results by considering numerical semigroups generalized by geometric sequences of the form $a^k, a^{k-1}b, \dots, b^k$, aiming to determine when the corresponding set of gaps is equidistributed modulo $m$. With elementary methods, we are able to obtain a result for $k=2$ and all $m$. We then work with cyclotomic rings, using results about multiplicative independence of cyclotomic units to obtain results for all $k$ and infinitely many $m$. Finally, we take an approach with cyclotomic units and Dirichlet L-functions to obtain results for all $k$ and all $m$. |
| title | Equidistribution Conditions for Gaps of Geometric Numerical Semigroups |
| topic | Number Theory 11D04, 20M14, 11F70, 11F66, 11R18, 11J82 |
| url | https://arxiv.org/abs/2503.10826 |