Explicit sumset sizes in additive number theory

Abstract

It is an open problem in additive number theory to compute and understand the full range of sumset sizes of finite sets of integers, that is, the set $\mathcal{R}_{\mathbf{Z}}(h,k)= \{|hA|:A \subseteq {\mathbf{Z}} \text{ and } |A|=k\}$ for all integers $h \geq 3$ and $k \geq 3$. This paper constructs certain infinite families of finite sets of size $k$ and computes their $h$-fold sumset sizes.