Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2604.24891 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Inhaltsangabe:
- 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 $A$.