Introduction

One general theme throughout this book is that sum sets A + B are more structured than arbitrary sets A, B, and in particular that iterated sum sets such as lA = {a1 + … + al : ai ∈ Ai} should get increasingly structured as l gets larger. One example of this phenomenon is Lemma 4.13, which shows that if A has small Fourier bias then l A quickly fills out the entire ambient group. (See also Exercise 4.3.12 for related demonstration of special structure of sum sets.) For another example, let A be a subset of [1, n] for some large n and consider (as a measure of structure) the longest progression contained inside A. If A has no structure other than density, e.g. ∣A∣ ≥ 0.99n, then there is not much we can say. Even the powerful quantitative version (11.23) of Szemerédi's theorem due to Gowers can only obtain an arithmetic progression of length Ω(log log log log log n). For cubes the situation is somewhat better (and simpler); Lemma 10.49 guarantees that A contains a proper cube of dimension Ω(log log n), though this is still far from the maximal dimension Θ(log n) of the cubes inside [1, n].

The situation improves markedly with taking sum sets, though. First, if A ⊂ [1, n] has cardinality at least 0.99n, then it is easy to see that A + A or A − A contains an arithmetic progression of length 0.98n.