  We investigate the R\'enyi entropy of independent sums of integer valued
random variables through Fourier theoretic means, and give sharp comparisons
between the variance and the R\'enyi entropy, for Poisson-Bernoulli variables.
As applications we prove that a discrete ``min-entropy power'' is super
additive on independent variables up to a universal constant, and give new
bounds on an entropic generalization of the Littlewood-Offord problem that are
sharp in the ``Poisson regime''.
