Theorem 19.2.537D86D761E81

Proof. The equalities (19.2.6) follow from theorem 19.7.20.

It remains to prove (19.2.7). By the argument of [Lemma 4.8, WN14], for any compact set $K\subseteq \operatorname{Int}\Delta (S)$, there is $k_0>0$ such that for any $k\geq k_0$, $\alpha \in K\cap (k^{-1}\mathbb {Z})^ n$ implies that $\alpha \in \Delta _ k(S)$.

To conclude, fix $\epsilon >0$. Choose $\delta >0$ such that $K\coloneqq \Delta (S)_{\delta }$ has positive volume and $d_{\mathrm{Haus}}(K,\Delta (S))<\epsilon /2$. Applying 19.7.7 to $K$, we find that $\Delta _ k(S)$ contains $K_{n^{1/2}k^{-1}}$ for all large $k$. After increasing $k$, we may also assume that $d_{\mathrm{Haus}}(K,K_{n^{1/2}k^{-1}})<\epsilon /2$. Since $\Delta _ k(S)\subseteq \Delta (S)$, this implies (19.2.7). $\square$