Kissing numbers and transference theorems from generalized tail bounds

TL;DR

This paper extends tail bounds for Gaussian lattice mass to broader functions, deriving new transference bounds and lattice point estimates, including bounds on kissing numbers in various norms.

Contribution

It generalizes tail bounds to a wider class of functions, leading to new transference bounds and lattice point estimates, notably in _p norms.

Findings

Bound the lattice kissing number in _p norms by e^{(n+ o(n))/p} for 0 < p  2

Established a new transference bound in the _1 norm

Generalized tail bounds for Gaussian lattice mass to broader test functions

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.