A strengthening of Freiman's 3k-4 theorem

TL;DR

This paper strengthens Freiman's 3k-4 theorem by showing that if the sumset of a subset with any four-element subset of another is small, then both sets are close to arithmetic progressions.

Contribution

It introduces a new condition involving sumsets with four-element subsets, extending the classical theorem's applicability.

Findings

Sets are close to arithmetic progressions under new sumset conditions

Small sumsets imply strong structural regularity

Generalizes Freiman's theorem to broader scenarios

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.