Sumsets with a minimum number of distinct terms

Abstract: For a set $A$ of $k$ elements from an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote this set by $h^{(\geq r)}A$. The set $h^{(\geq r)}A$ generalizes the classical sumsets $hA$ and $h\hat{}A$ for $r=1$ and $r=h$, respectively. As the main result of this article, we give an upper bound for the minimum size of $h^{(\geq r)}A$ over $\mathbb{Z}_m$ for $m \geq 2$. Further, by an observation relating the sumsets $hA$, $h\hat{}A$, and $h^{(\geq r)}A$ we obtain the sharp lower bound on the size of $h^{(\geq r)}A$ and also characterize the set $A$ for which the lower bound on the size of $h^{(\geq r)}A$ is tight over the groups $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p$ is a prime number.