Paper 2018/188

Generalizations of Banaszczyk's transference theorems and tail bound

Stephen D. Miller and Noah Stephens-Davidowitz

Abstract

We generalize Banaszczyk's seminal tail bound for the Gaussian mass of a lattice to a wide class of test functions. We therefore obtain quite general transference bounds, as well as bounds on the number of lattice points contained in certain bodies. As examples, we bound the lattice kissing number in $\ell_p$ norms by $e^{(n+o(n))/p}$ for $0 < p \leq 2$, and also give a proof of a new transference bound in the $\ell_1$ norm.

Note: Preprint, 18 pages.