PomeCorollary 19.9.7.071721B08FC2 If the subgroup generated by $S$ is all of $\mathbb Z^{n+1}$, then
\[ \lim _{m\to \infty }m^{-n}\dim W_ m =\operatorname {vol}_{\mathbb R^ n}\Delta (W_\bullet ). \]
Corollary 19.9.7071721B08FC2
Proof. The semigroup $S$ satisfies the cone conditions established above and, by hypothesis, generates $\mathbb Z^{n+1}$. The semigroup counting theorem therefore applies to it. By lemma 19.9.2, the number of points of $S$ at height $m$ is $\dim W_ m$, and the height-one slice of the closed convex cone generated by $S$ is $\Delta (W_\bullet )$. $\square$
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