Inverse problems for sumset sizes of finite sets of integers

Let $A$ be a finite set of integers and let $hA$ be its $h$-fold sumset. This paper investigates the sequence of sumset sizes $(\mathrm{\mid }hA\mathrm{\mid }{)}_{h=1}^{\mathrm{\infty }}$, the relations between these sequences for affinely inequivalent sets $A$ and $B$, and the comparative growth rates and configurations of the sumset size sequences $(\mathrm{\mid }hA\mathrm{\mid }{)}_{h=1}^{\mathrm{\infty }}$ and $(\mathrm{\mid }hA\mathrm{\mid }{)}_{h=1}^{\mathrm{\infty }}$.