  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.
