Gap sets of random generalized numerical semigroups

For a fixed positive integer $d$ and a small real $p>0$, sample a $p$-random subset $A\subseteq {\mathbb{Z}}_{\ge 0}^d$, and let $S:=⟨A⟩$ be the generalized numerical semigroup generated by $A$. We show that with high probability (as $p\to 0$), the gap set ${\mathbb{Z}}_{\ge 0}^d\setminus S$ is well approximated by the shifted hyperboloid region $$\{(x_1,\dots ,x_d)\in {\mathbb{R}}_{\ge 0}^d:(x_1+\log p^{-1})\cdots (x_d+\log p^{-1})\ll p^{-1}(\log p^{-1}{)}^{d+1}\}\mathrm{.}$$ 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 $A$.