Lattice-point asymptotics by fundamental cells ($N(R)=\mathrm{vol}(B_1)R^d/A+O(R^{d-1})$)

For a full-rank Euclidean lattice of covolume $A$ in ${\mathbb{R}}^d$, bounded fundamental cells compare lattice points in a radius-$R$ ball with the volumes of balls of radii $R\pm C$. Hence $N(R)=\mathrm{vol}(B_1)R^d/A+O(R^{d-1})$. A lattice norm multiset therefore determines the covolume, which is useful for the spectrum of a flat torus.