Kissing Numbers and Transference Theorems from Generalized Tail Bounds

Abstract

We generalize Banaszczyk's seminal tail bound for the Gaussian mass of a lattice to a wide class of test functions. From this we obtain quite general transference bounds, as well as bounds on the number of lattice points contained in certain bodies. As applications, 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.