A simple proof of a reverse Minkowski theorem for integral lattices

Abstract

We prove that for any integral lattice $\mathcal{L}\subset {\mathbb{R}}^n$ (that is, a lattice $\mathcal{L}$ such that the inner product $⟨{\mathbf{y}}_1,{\mathbf{y}}_2⟩$ is an integer for all ${\mathbf{y}}_1,{\mathbf{y}}_2\in \mathcal{L}$) and any positive integer $k$, \[ |\{ \mathbf{y} \in \mathcal{L} \ : \ \|\mathbf{y}\|^2 = k\}| \leq 2 \binom{n+2k-2}{2k-1} \; , \] giving a nearly tight reverse Minkowski theorem for integral lattices.