Plunnecke's inequality for different summands

TL;DR

This paper generalizes Pl"unnecke's inequality to multiple summands, providing a method to find subsets with controlled sumset sizes, and applies it to extend submultiplicativity inequalities.

Contribution

It introduces a generalized version of Pl"unnecke's inequality for multiple summands and demonstrates its application to sumset submultiplicativity.

Findings

Existence of subsets with controlled sumset sizes

Generalization of submultiplicativity inequality

Applicable to multiple summands in additive combinatorics

Abstract

The aim of this paper is to prove a general version of Pl\"unnecke's inequality. Namely, assume that for finite sets $A$, $B_1, ... B_k$ we have information on the size of the sumsets $A+B_{i_1}+... +B_{i_l}$ for all choices of indices $i_1, ... i_l.$ Then we prove the existence of a non-empty subset $X$ of $A$ such that we have `good control' over the size of the sumset $X+B_1+... +B_k$. As an application of this result we generalize an inequality of \cite{gymr} concerning the submultiplicativity of cardinalities of sumsets.