A problem on sumset sizes of sets of lattice points

A central problem in additive number theory is to understand the set of sizes of $h$-fold sums of finite subsets of an additive abelian semigroup.

It is proved that the "range of sumset sizes'' is the same for finite sets of integers and for finite sets of $n$-dimensional lattice points.

The associated geometrical problem is to determine if these sumset size sets can be computed more efficiently with lattice points than with integers.