Plunnecke's inequality for different summands

Abstract

The aim of this paper is to prove a general version of Plünnecke'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.

8 pages