A Reverse Minkowski Theorem

TL;DR

This paper proves a conjecture relating to lattices with certain determinant properties, providing bounds on short vectors and the covering radius, thus offering a partial converse to Minkowski's theorem.

Contribution

It establishes a new bound on the theta sum for lattices with sublattice determinants at least one, confirming a conjecture and deriving bounds on short vectors and covering radius.

Findings

Bound on the theta sum for specific lattices.

Bounds on the number of short lattice vectors.

Bound on the covering radius.

Abstract

$ \newcommand{\R}{\mathbb{R}} \newcommand{\lat}{\mathcal{L}} $We prove a conjecture due to Dadush, showing that if $\lat \subset \R^n$ is a lattice such that $\det(\lat') \ge 1$ for all sublattices $\lat' \subseteq \lat$, then \[ \sum_{\vec y \in \lat} e^{-\pi t^2 \|\vec y\|^2} \le 3/2 \; , \] where $t := 10(\log n + 2)$. From this we derive bounds on the number of short lattice vectors, which can be viewed as a partial converse to Minkowski's celebrated first theorem. We also derive a bound on the covering radius.