A well known result about the natural numbers $\mathbb{N}$ says that for any finite subset $A \subset \mathbb{N}$ there exists $R \ge 0$ such that if $n$ is in the subgroup of $\mathbb{Z}$ generated by $A$ and if $n \ge R$ then $n$ is in the semigroup generated by $A$.

Are there any references to a higher dimensional version of this result?

The version I want goes like this.

• Take a finite subset $U$ of $\mathbb{N}^n$. Let $C_U$ be the smallest closed cone in $\mathbb{R}^n$ containing $U$, i.e. all non-negative real linear combinations of $U$. Let $G_U$ be the subgroup of $\mathbb{N}^n$ generated by $U$, i.e. all integer linear combinations. Let $S_U$ be the subsemigroup generated by $U$, i.e. all non-negative integer linear combinations. Then there exists $R>0$ such that for every $v \in G_U$, if the ball around $v$ of radius $R$ is contained in $C_U$ then $v \in S_U$.