$ \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 \[ \sum_{\vec y \in \lat} e^{-t^2 \|\vec y\|^2} \le 3/2 \; , \] where $t := 10(\log n + 2)$. From this we also derive bounds on the number of short lattice vectors and on the covering radius.
['Oded Regev', 'Noah Stephens-Davidowitz']
A Reverse Minkowski Theorem