# An improved constant in Banaszczyk's transference theorem

## TL;DR

This paper improves the constant factor in Banaszczyk's transference theorem relating lattice covering radius and dual lattice shortest vector, using a packing-based bound instead of Fourier analysis.

## Contribution

It introduces a new proof step replacing Fourier bounds with packing bounds, achieving a 20% improvement in the transference constant.

## Key findings

- Improved the constant in Banaszczyk's transference theorem by about 20%.
- Replaced Fourier-analytic bounds with packing-based bounds in the proof.
- Demonstrated that existing packing bounds can enhance classical lattice theorems.

## Abstract

$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that \[ \mu(\lat) \lambda_1(\lat^*) < \big( 0.1275 + o(1) \big) \cdot n \; , \] where $\mu(\lat)$ is the covering radius of an $n$-dimensional lattice $\lat \subset \R^n$ and $\lambda_1(\lat^*)$ 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.