A reverse Minkowski theorem

Abstract

We prove a conjecture due to Dadush, showing that if $\mathcal{L} \subset \mathbb{R}^n$ is a lattice such that $\mathrm{det}(\mathcal{L}’)\ge 1$ for all sublattices $\mathcal{L}’ \subseteq \mathcal{L}$, then $\sum_{\mathbf{y}\in \mathcal{L}} e^{-\pi t^2 \|\mathbf{y} \|^2} \le 3/2$, 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.