For a lattice $L\subset \mathbb{R}^d$ with base region $B$, how many points are in $L\cap nB$?

Question

Take a lattice $L$ in $\mathbb{R}^d$ and a convex region $B$ containing $0$ so that $\{B+\ell:\ell \in L\}$ partitions $\mathbb{R}^d$. Does $nB$ necessarily contain $n^d$ points in $L$? Convexity might not be needed.

Answer 1

The map $n\mathbf{I}_{d\times d}$ turns group $L$ into subgroup $nL\subset L$. Then by Corrollary 8.2.1 of Lattice Coding of Signals and Networks by Zamir, $|nB\cap L| = |\operatorname{det} n\mathbf{I}_{d\times d}|=n^d$

Proof is given by a somewhat involved discussion surrounding it.