Compression and complexity for sumset sizes in additive number theory

Abstract: 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 $B_h$-sets). This paper considers the sets ${\mathcal R}{\mathbf Z}(h,k)$ and ${\mathcal R}{{\mathbf Z}^n}(h,k)$ 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 ${\mathcal R}{\mathbf Z}(h,k)$ and ${\mathcal R}{{\mathbf Z}^n}(h,k)$.