Compression, Complements and the 3k−4 Theorem

Abstract

We have seen (from Kneser’s Theorem) that if |A+B|<|A|+|B|−1, then A+B must be periodic, which may be viewed as a partial structural description of A+B. In this chapter, we will prove a more precise structure theorem describing $A,\,B\subseteq \mathbb{Z}$ with

$$|A+B|\leq |A|+|B|-3+\min\{|B|-\delta(A,B),\,|A|-\delta(B,A)\}, $$

where

$$\delta(A,B)=\left\{ \begin{array}{ll} 1 & \mbox{if } x+A\subseteq B \mbox{ for some } x\in \mathbb{Z}\mbox{,} \\ 0 & \mbox{otherwise,} \end{array} \right. $$

which shows that subsets of very small sumset in $\mathbb{Z}$ must be large subsets of a pair of arithmetic progressions with common difference and their sumset must contain a large arithmetic progression of this same difference. Along the way, we will introduce the notion of relative complements and dual pairs, which we will need later in the course. Compression techniques and the related method of modular reduction will be developed in very general form and (as a simple example of compression techniques) we will derive a discrete Brunn-Minkowski Theorem in dimension 2.