Let ${\bf A} = (A_1 , \ldots , A_q )$ be a $q$-tuple of finite sets of integers. Associated to every $q$-tuple of nonnegative integers ${\bf h} = (h_1 , \ldots , h_q )$ is the linear form ${\bf h}\cdot {\bf A} = h_1 A_1 + \ldots + h_q A_q$ . The set $({\bf h}\cdot {\bf A})^{(t)}$ consists of all elements of this sumset with at least $t$ representations. The structure of the set $({\bf h}\cdot {\bf A})^{(t)}$ is computed for all sufficiently large $h_i$ .
Mel Nathanson