New Proofs of Pl\"unnecke-type Estimates for Product Sets in Groups

TL;DR

This paper introduces a novel method for bounding the size of product sets in groups, providing new proofs of key inequalities and extending results to non-commutative groups.

Contribution

It offers new, concise proofs of Plunnecke-Ruzsa inequalities and Tao's theorem, along with generalizations to non-Abelian groups.

Findings

Short proof of Plunnecke-Ruzsa inequalities for Abelian groups

New proof of Tao's theorem on triple products

Generalization of inequalities to non-commutative groups

Abstract

We present a new method to bound the cardinality of triple product sets in groups and give three applications. A new and unexpectedly short proof of the Plunnecke-Ruzsa sumset inequalities for Abelian groups. A new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made. A further generalisation of the Plunnecke-Ruzsa inequalities in general groups.