One can prove that for $A$ and $B$ subsets of $\mathbb{Z}$ one has $$ |A|+|B|-1 \leq |A+B| \leq |A|\times |B| $$ where $A+B = \left\{ a+b,a\in A \text{ and } b \in B \right\}$ and, for $X$ a finite set, $|X|$ denotes its cardinal. I'm not interested on a proof of the two previous inequalities but the optimality of these inequalities in the sense on the following exercise.
I've just found here the following exercise (that proves that the previous bound are tight in general) : for $m,n,s$ integers such that $m\geq 1$, $n\geq 1$ and $m+n-1 \leq s \leq m\times n$, we can find $A$ and $B$ subsets of $\mathbb{Z}$ such that $|A|=m$, $B=n$ and $|A+B|=s$.
Do you have any hint for this exercise ?