A multiple set version of the 3k -3 Theorem

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 A such that |A + A| ≤ 3|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 A such that the inequality |jA| ≤ j(j + 1)(|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 jA.