A multiple set version of the $3k-3$ Theorem

Abstract

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 $|\mathcal{A}+\mathcal{A}|\le 3|\mathcal{A}|-3$. 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 $|j\mathcal{A}|\le j(j+1)(|\mathcal{A}|-1)/2$ 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 $j\mathcal{A}$.