PomeTheorem 19.7.20 (Kaveh–Khovanskii).85641516C8CA Let $S\in \mathcal{S}(C)$. Then, for every sufficiently divisible positive integer $a$, the limit
\[ \lim _{k\to \infty }(ka)^{-n}|S_{ka}| \]
exists, is independent of $a$, and is equal to $\varlimsup _{k\to \infty } k^{-n}|S_ k|$. If moreover $S$ generates $\mathbb {Z}^{n+1}$ as an Abelian group, and if $C(S)$ denotes the closed convex cone generated by $S\cup \{ 0\} $, then
\[ \lim _{k\to \infty } k^{-n}|S_ k| = \operatorname{vol}\{ x\in \mathbb {R}^ n:(x,1)\in C(S)\} . \]
In particular, for every convex body $K\subseteq \mathbb {R}^ n$,
\[ \lim _{k\to \infty } k^{-n}\# \left(kK\cap \mathbb {Z}^ n\right)=\operatorname{vol}K. \]
The same assertion holds with $\mathbb {Z}^ n$ replaced by any lattice, provided the Lebesgue measure is normalized by that lattice.
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