Statistics of Erdős-Rényi random numerical semigroups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909791218040832 |
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| author | Kravitz, Noah Morales, Santiago Schildkraut, Carl |
| author_facet | Kravitz, Noah Morales, Santiago Schildkraut, Carl |
| contents | For $p>0$ a small parameter, let $\mathcal A \subseteq \mathbb{Z}_{>0}$ be a random subset where each positive integer is included independently with probability $p$. We show that, with high probability (as $p \to 0$), the numerical semigroup $\langle\mathcal A\rangle:=\{a_1+\cdots+a_k: k \geq 0, a_1, \ldots, a_k \in \mathcal A\}$ generated by $\mathcal A$ has Frobenius number and genus of size $\asymp p^{-1}(\log p^{-1})^2$ and embedding dimension of size $\asymp (\log p^{-1})^2$. This resolves an open problem of Bogart and the second author. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistics of Erdős-Rényi random numerical semigroups Kravitz, Noah Morales, Santiago Schildkraut, Carl Combinatorics Commutative Algebra Number Theory For $p>0$ a small parameter, let $\mathcal A \subseteq \mathbb{Z}_{>0}$ be a random subset where each positive integer is included independently with probability $p$. We show that, with high probability (as $p \to 0$), the numerical semigroup $\langle\mathcal A\rangle:=\{a_1+\cdots+a_k: k \geq 0, a_1, \ldots, a_k \in \mathcal A\}$ generated by $\mathcal A$ has Frobenius number and genus of size $\asymp p^{-1}(\log p^{-1})^2$ and embedding dimension of size $\asymp (\log p^{-1})^2$. This resolves an open problem of Bogart and the second author. |
| title | Statistics of Erdős-Rényi random numerical semigroups |
| topic | Combinatorics Commutative Algebra Number Theory |
| url | https://arxiv.org/abs/2509.12862 |