As 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 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 $|A_1 + \cdots + A_{i}| = |A_1| + \cdots + |A_i|$.