Compression and complexity for sumset sizes in additive number theory

The study of sums of finite sets of integers has mostly concentrated on sets with small sumsets (Freiman's theorem and related work) and on sets with large sumsets (Sidon sets and Bₕ-sets). This paper considers the sets RZ(h,k) and R_ Zⁿ(h,k) of all sizes of h-fold sums of sets of k integers or of k lattice points, and the geometric and computational complexity of the sets RZ(h,k) and R_ Zⁿ(h,k). For sumsets $hA$ with large diameter, there is a compression algorithm to construct sets $A'$ with $|hA'| = |hA|$ and small diameter.