Lattice-point asymptotics by fundamental cells
= Lattice-point asymptotics by fundamental cells
{title2=$N(R)=\operatorname{vol}(B_1)R^d/A+O(R^{d-1})$}
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 $N(R)=\operatorname{vol}(B_1)R^d/A+O(R^{d-1})$. A lattice norm multiset therefore determines the covolume, which is useful for the <spectrum of a flat torus>.