A Note on Sumsets and Restricted Sumsets

Abstract

In this note we find the optimal lower bound for the size of the sumsets $HA$ and $H\hat{}A$ over finite sets $H,A$ of nonnegative integers, where $HA={\bigcup }_{h\in H}hA$ and $H\hat{}A={\bigcup }_{h\in H}h\hat{}A$. We also find the underlying algebraic structure of the sets $A$ and $H$ for which the size of the sumsets $HA$ and $H\hat{}A$ is minimum.

8 pages