The Cardinality of Sumsets: Different Summands

Abstract

Let $h$ be a positive integer and $A,B_1,B_2,\dots ,B_h$ be finite sets in a commutative group. We bound $|A+B_1+...+B_h|$ from above in terms of $|A|,|A+B_1|,\dots ,|A+B_h|$ and $h$. Extremal examples, which demonstrate that the bound is asymptotically sharp in all the parameters, are furthermore provided.

25 pages. A minor change in the statement of Lemma 5.13