In 1959, Freiman demonstrated his famous $3k-4$ Theorem which was
to be a cornerstone in inverse additive number theory. This result
was soon followed by a $3k-3$ Theorem, proved again by Freiman.
This result describes the sets of integers $\mathcal{A}$ such that
$| \mathcal{A}+\mathcal{A}
| \leq 3 | \mathcal{A} | -3$. In the present paper, we prove a $3k-3$-like
Theorem in the context of multiple set addition and describe, for
any positive integer $j$, the sets of integers $\mathcal{A}$ such that the
inequality $|j \mathcal{A} | \leq j(j+1)(| \mathcal{A} | -1)/2$ holds. Freiman's
$3k-3$ Theorem is the special case $j=2$ of our result. This
result implies, for example, the best known results on a function
related to the Diophantine Frobenius number. Actually, our main
theorem follows from a more general result on the border of $j\mathcal{A}$.