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

TL;DR

The paper extends the Littlewood-Offord problem to independent discrete log-concave random variables, establishing a concentration bound in terms of total variance and giving a corresponding entropic bound via Rényi entropy power. It develops a sign-majorization framework to reduce the problem to signed Bernoulli-type cases and proves both variance- and entropy-based inequalities, including a LO-type bound for arithmetic progressions. Special cases for Bernoulli and discrete uniform distributions are analyzed, yielding sharpened bounds and a discrete-uniform entropy-power inequality that unify and extend prior results. The work thus advances the understanding of concentration phenomena for sums of discrete log-concave variables and provides discrete information-theoretic tools with potential applications in related combinatorial and probabilistic contexts.

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.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: A, BMM
• Lemma 2.2
• proof
• Theorem 2.3: MWW
• Theorem 3.1
• proof