A simple proof of a reverse Minkowski theorem for integral lattices

Abstract

We prove that for any integral lattice $L\subset R^n$ (that is, a lattice $L$ such that the inner product $〈y_1,y_2〉$ is an integer for all $y_1,y_2\in L$) and any positive integer k,$\left|\right\{y\in L:{\Vert y\Vert }^2=k\left\}\right|\le 2\left(\begin{array}{c}n+2k-2\\ 2k-1\end{array}\right),$ giving a nearly tight reverse Minkowski theorem for integral lattices.