Gap sets of random generalized numerical semigroups

TL;DR

This paper studies the structure of gaps in random generalized numerical semigroups in multiple dimensions, showing they are well approximated by a shifted hyperboloid region as the sampling probability approaches zero.

Contribution

It generalizes previous one-dimensional results to higher dimensions, providing a probabilistic approximation of the gap set in random generalized numerical semigroups.

Findings

The gap set is approximated by a shifted hyperboloid region with high probability.

The results extend to the set of subset sums of the generating set.

The approximation holds as the sampling probability p approaches zero.

Abstract

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 $A$.