A Reverse Minkowski Theorem

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.