Bernoulli sums and Rényi entropy inequalities

We investigate the Rényi entropy of sums independent integer-valued random variables through Fourier theoretic means, and give sharp comparisons between variance for Bernoulli variables. As applications, we prove that a discrete “min-entropy power” is superadditive with respect to convolution modulo universal constant, new bounds on an entropic generalization Littlewood-Offord problem are in “Poisson regime”.