A Reverse Minkowski Theorem

\newcommand{\R}{\mathbb{R}} \newcommand{\lat}{\mathcal{L}}We prove a conjecture due to Dadush, showing that if $\text{lat}\subset {\mathbb{R}}^n$ is a lattice such that $\det ({\text{lat}}^{\mathrm{\prime }})\ge 1$ for all sublattices ${\text{lat}}^{\mathrm{\prime }}\subseteq \text{lat}$, then $$\sum _{\vec{y}\in \text{lat}}{e}^{-\pi t^2\mathrm{\parallel }\vec{y}{\mathrm{\parallel }}^2}\le 3\mathrm{/}2,$$ 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.