Entropy Inequalities for Sums in Prime Cyclic Groups

TL;DR

This paper establishes lower bounds for Rényi entropies of sums of independent variables in prime cyclic groups, using rearrangement inequalities and stochastic ordering, with applications to entropy power inequalities and combinatorial problems.

Contribution

It introduces new entropy inequalities in prime cyclic groups based on extended rearrangement techniques and stochastic orderings, expanding the theoretical understanding of entropy in discrete groups.

Findings

Derived lower bounds for Rényi entropies of sums in prime cyclic groups.

Applied inequalities to discrete entropy power inequalities and combinatorial problems.

Extended rearrangement inequalities to prime cyclic groups.

Abstract

Lower bounds for the R\'enyi 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 (2001), 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.