The Cardinality of Sumsets: Different Summands

TL;DR

This paper establishes upper bounds on the size of sumsets involving multiple summands in a commutative group, providing extremal examples that show the bounds are nearly optimal across all parameters.

Contribution

It introduces new bounds for the cardinality of sumsets with multiple summands and constructs extremal examples demonstrating the bounds' sharpness.

Findings

Derived upper bounds for |A+B_1+...+B_h| in terms of individual set sizes.

Provided extremal examples confirming the bounds are asymptotically sharp.

Extended understanding of sumset cardinalities in additive combinatorics.

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.