A note on the Littlewood-Offord problem for discrete log-concave distributions

TL;DR

This paper extends the Littlewood-Offord problem to discrete log-concave distributions, exploring variants for arithmetic progressions and entropy, and generalizes entropy power inequalities for discrete uniform distributions.

Contribution

It introduces an extension of the Littlewood-Offord problem for discrete log-concave distributions and discusses related entropy and arithmetic progression variants.

Findings

Extended Littlewood-Offord problem to discrete log-concave distributions

Developed entropy and arithmetic progression variants

Generalized entropy power inequality for discrete uniform distributions

Abstract

We present an extension of the famous Littlewood-Offord problem when Bernoulli distributions are replaced with discrete log-concave distributions. A variant of the Littlewood-Offord problem for arithmetic progressions, as well as an entropic version, is also discussed. Along the way, we recover and extend a result of Madiman and Woo (2015) on the entropy power inequality for discrete uniform distributions.