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

• Published in: The Bulletin of the London Mathematical Society 2023-10, Vol.55 (5), p.2363-2381
• Main Authors: Bollobás, Béla, Leader, Imre, Tiba, Marius
• Format: Article
• Language: English
• ISSN: 0024-6093; 1469-2120
• DOI: 10.1112/blms.12862

In its usual form, Freiman's 3k−4$3k-4$ theorem states that if A$A$ and B$B$ are subsets of Z${\mathbb {Z}}$ of size k$k$ with small sumset (of size close to 2k$2k$), 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 A$A$ and B$B$ are subsets of Z${\mathbb {Z}}$ of size k$k$ such that for any four‐element subset X$X$ of B$B$ the sumset A+X$A+X$ has size not much more than 2k$2k$, then already this implies that A$A$ and B$B$ are very close to arithmetic progressions.