| Popis: |
In its usual form, Freiman's 3 k − 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 2 k $2k$ ), 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 2 k $2k$ , then already this implies that A $A$ and B $B$ are very close to arithmetic progressions. |
