It is trivial that $|A\_1 + \cdots + A\_h| \leq |A\_1|\cdots |A\_h|$, where $h \geq 2$ and $A\_i \subseteq \mathbb{Z}$ are nonempty finite sets and $A\_1 + \cdots + A\_h :=\{a\_1 + \cdots + a\_h : a\_i \in A\_i ~\text{for}~ i = 1, \ldots, h\}$. I am interested in finding those sets for which $|A\_1 + \cdots + A\_h|$ is exactly $|A\_1|\cdots |A\_h|$. I have some specific examples for the case $h = 2$. 



Sumset achieving extreme upper bound
Elaborating on what Gerhard Paseman points out, selecting sets $A\_i = \{kh^i \mid 0 \le k < h\}$ will work.


More generally, if you define your sets inductively (and restrict to only using positive integers) so that the smallest pairwise difference in elements in $A\_i$ is greater than the largest element in $A\_1 + \cdots + A\_{i-1}$ then it will also avoid any collisions, and have $|A\_1 + \cdots + A\_{i}| = |A\_1| \cdots |A\_i|$.


Edit to provide example:


For example, choosing $A\_1 = \{0,1,4\}$ and then $A\_2$ any set with elements at least $5$ apart $A\_2 = \{0,7, 22, 54, 59\}$ and now $A\_3$ any set with elements at least $64$ apart $A\_3 = \{0, 77, 200\}$ and $A\_4$ having elements at least $264$ apart, etc. You can continue at will.


