Pl\"unnecke's Inequality

TL;DR

The paper presents a new proof of Plünnecke's inequality, a fundamental result in additive combinatorics, avoiding traditional graph-theoretic tools and exploring the conditions for the inequality's sharpness.

Contribution

It introduces a novel proof technique for Plünnecke's inequality and analyzes the sharpness and attainability conditions of the inequality.

Findings

New proof of Plünnecke's inequality without Menger's theorem

Sharpness can be achieved for arbitrarily long graphs

Necessary conditions for the inequality to be tight

Abstract

Plunnecke's inequality is the standard tool to obtain estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the proof is completed with no reference to Menger's theorem or Cartesian products of graphs. We also investigate the sharpness of the inequality and show that it can be sharp for arbitrarily long, but not for infinite commutative graphs. A key step in our investigation is the construction of arbitrarily long regular commutative graphs. Lastly we prove a necessary condition for the inequality to be attained.