From an entropic inequality for sums in abelian groups to the theorems of Ruzsa and Plünnecke

2026-10-08

AI Summary

This study addresses the Plünnecke–Ruzsa inequality for subsets of finite abelian groups, whose traditional proofs rely on tensor powers and asymptotic constructions that are intricate and difficult to generalize. To overcome these limitations, this work proposes a direct information-theoretic proof framework based on joint-entropy-maximizing coupling inequalities, thereby circumventing conventional asymptotic analysis. The proposed approach yields elementary and streamlined derivations of Ruzsa’s theorem, Plünnecke’s theorem, and the triangle inequality. Furthermore, it successfully extends the theoretical scope of these results to non-abelian groups. By replacing complex combinatorial machinery with concise information-theoretic arguments, this research establishes an elegant new paradigm for additive combinatorics.

Abstract

We give a direct information-theoretic proof of the Plünnecke--Ruzsa inequalities for finite subsets of abelian groups. The argument rests on an elementary coupling inequality proved by maximizing joint entropy, without tensor powers or asymptotic constructions. It also yields Ruzsa's sumset triangle inequality and an extension to non-abelian groups.

Problem

• Plünnecke-Ruzsa inequalities
• abelian groups
• information-theoretic proof
• sumset triangle inequality
• entropic inequality

Innovation

• entropic inequality
• Plünnecke-Ruzsa inequalities
• information-theoretic proof
• joint entropy maximization
• sumset triangle inequality