A strengthening of Freiman's 3k−4$3k-4$ theorem

In its usual form, Freiman's theorem states that if and are subsets of of size with small sumset (of size close to ), then they are very close to arithmetic progressions. Our aim in this paper is to strengthen this by allowing only a bounded number of possible summands from one of the sets. We show that if and are subsets of of size such that for any four‐element subset of the sumset has size not much more than , then already this implies that and are very close to arithmetic progressions.