A simple proof of a reverse Minkowski theorem for integral lattices

TL;DR

This paper presents a straightforward proof of a nearly tight reverse Minkowski theorem for integral lattices, providing bounds on the number of lattice vectors of a given squared norm.

Contribution

The paper offers a simple, elegant proof of a reverse Minkowski theorem specifically for integral lattices, establishing nearly tight bounds.

Findings

Bounds on the number of vectors with a fixed squared norm in integral lattices

A nearly tight reverse Minkowski theorem for integral lattices

Simplified proof technique for lattice norm enumeration

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 $\langle \mathbf{y}_1,\mathbf{y}_2 \rangle$ 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.