An improved constant in Banaszczyk's transference theorem

$ \newcommand{\R}{\ensuremath{$\mathbb{R}$}} \newcommand{\lat}{$\mathcal{L}$} \newcommand{\ensuremath}[1]{#1} $We show that $$\mu (\text{lat}){\lambda }_1({\text{lat}}^{\ast })<(0.1275+o(1))\cdot n,$$ where $\mu (\text{lat})$ is the covering radius of an $n$-dimensional lattice $\text{lat}\subset {\mathbb{R}}^n$ and ${\lambda }_1({\text{lat}}^{\ast })$ is the length of the shortest non-zero vector in the dual lattice ${\text{lat}}^{\ast }$. 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.