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 article constructs certain infinite families of finite sets of size k , computes their h -fold sumset sizes, and obtains explicit finite arithmetic progressions of sumset sizes in $ \mathcal R_{\mathbf Z}(h,k)$ .