Entropy Inequalities for Sums in Prime Cyclic Groups

Lower bounds for the Rényi entropies of sums of independent random variables taking values in cyclic groups of prime order under permutations are established. The main ingredients of our approach are extended rearrangement inequalities in prime cyclic groups building on Lev [Duke Math. J., 107 (2001), pp. 239--263] and notions of stochastic ordering. Several applications are developed, including to discrete entropy power inequalities, the Littlewood--Offord problem, and counting solutions of certain linear systems.