Result: A multiple set version of the $3k-3$ Theorem: 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 \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} .