A single set improvement to the 3k − 4 theorem


The 3k−4 Theorem is a classical result which asserts that if A,B⊆Z are finite, nonempty subsets with |A+B|=|A|+|B|+r≤|A|+|B|+min⁡{|A|,|B|}−3−δ, where δ=1 if A and B are translates of each other, and otherwise δ=0, then there are arithmetic progressions PA and PB of common difference such that A⊆PA, B⊆PB, |B|≤|PB|+r+1 and |PA|≤|A|+r+1. It is one of the few cases in Freiman's Theorem for which exact bounds on the sizes of the progressions are known. The hypothesis (1) is best possible in the sense that there are examples of sumsets A+B having cardinality just one more than that of (1), yet A and B cannot both be contained in short length arithmetic progressions. In this paper, we show that the hypothesis (1) can be significantly weakened and still yield the same conclusion for one of the sets A and B. Specifically, if |B|≥3, s≥1 is the unique integer with (s−1)s([Formula presented] and [Formula presented] then we show there is an arithmetic progression PB⊆Z with B⊆PB and |PB|≤|B|+r+1. The hypothesis (2) is best possible (without additional assumptions on A) for obtaining such a conclusion.

Publication Title

Journal of Number Theory