We investigate the Renyi entropy of independent sums of integer valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Renyi 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.
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