A strengthening of Freiman's 3k-4 theorem
Combinatorics
2022-04-22 v1
Abstract
In its usual form, Freiman's 3k-4 theorem states that if A and B are subsets of the integers of size k with small sumset (of size close to 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 and B are subsets of the integers of size k such that for any four-element subset X of B the sumset A+X has size not much more than 2k then already this implies that A and B are very close to arithmetic progressions.
Cite
@article{arxiv.2204.09816,
title = {A strengthening of Freiman's 3k-4 theorem},
author = {Bela Bollobas and Imre Leader and Marius Tiba},
journal= {arXiv preprint arXiv:2204.09816},
year = {2022}
}