An improved constant in Banaszczyk's transference theorem

Abstract

$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that \[ μ(\lat) λ_1(\lat^*) < \big( 0.1275 + o(1) \big) \cdot n \; , \] where $μ(\lat)$ is the covering radius of an $n$-dimensional lattice $\lat \subset \R^n$ and $λ_1(\lat^*)$ is the length of the shortest non-zero vector in the dual lattice $\lat^*$. This improves on Banaszczyk's celebrated transference theorem (Math. Annal., 1993) by about 20%. Our proof follows Banaszczyk exactly, except in one step, where we replace a Fourier-analytic bound on the discrete Gaussian mass with a slightly stronger bound based on packing. The packing-based bound that we use was already proven by Aggarwal, Dadush, Regev, and Stephens-Davidowitz (STOC, 2015) in a very different context. Our contribution is therefore simply the observation that this implies a better transference theorem.