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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2604.24891 |
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| _version_ | 1866914512649584640 |
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| author | Bitonti, Veronica Kravitz, Noah |
| author_facet | Bitonti, Veronica Kravitz, Noah |
| contents | For a fixed positive integer $d$ and a small real $p>0$, sample a $p$-random subset $A \subseteq \mathbb{Z}_{\geq 0}^d$, and let $S:=\langle A \rangle$ be the generalized numerical semigroup generated by $A$. We show that with high probability (as $p \to 0$), the gap set $\mathbb{Z}_{\geq 0}^d \setminus S$ is well approximated by the shifted hyperboloid region $$\{(x_1, \ldots, x_d) \in \mathbb{R}_{\geq 0}^d: (x_1+\log p^{-1}) \cdots (x_d+\log p^{-1})\ll p^{-1}(\log p^{-1})^{d+1}\}.$$ This generalizes work of the second author, Morales, and Schildkraut on the $1$-dimensional setting. We also obtain the same result with $S$ replaced by the set of subset sums of $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24891 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gap sets of random generalized numerical semigroups Bitonti, Veronica Kravitz, Noah Combinatorics For a fixed positive integer $d$ and a small real $p>0$, sample a $p$-random subset $A \subseteq \mathbb{Z}_{\geq 0}^d$, and let $S:=\langle A \rangle$ be the generalized numerical semigroup generated by $A$. We show that with high probability (as $p \to 0$), the gap set $\mathbb{Z}_{\geq 0}^d \setminus S$ is well approximated by the shifted hyperboloid region $$\{(x_1, \ldots, x_d) \in \mathbb{R}_{\geq 0}^d: (x_1+\log p^{-1}) \cdots (x_d+\log p^{-1})\ll p^{-1}(\log p^{-1})^{d+1}\}.$$ This generalizes work of the second author, Morales, and Schildkraut on the $1$-dimensional setting. We also obtain the same result with $S$ replaced by the set of subset sums of $A$. |
| title | Gap sets of random generalized numerical semigroups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.24891 |