Bernoulli sums and R\'enyi entropy inequalities

TL;DR

This paper explores the relationship between Renyi entropy and variance for sums of independent integer-valued variables, providing sharp inequalities and applications to entropy power and Littlewood-Offord problems.

Contribution

It introduces Fourier-based methods to compare Renyi entropy and variance, establishing super additivity of discrete min-entropy power and sharp bounds in the Poisson regime.

Findings

Sharp inequalities between Renyi entropy and variance for Poisson-Bernoulli sums

Proof of super additivity of discrete min-entropy power up to a universal constant

New bounds on entropic Littlewood-Offord problem in the Poisson regime

Abstract

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''.