Bourgain-Chang's proof of the weak Erdős-Szemerédi conjecture

Abstract

This is an exposition of the following `weak' Erdős-Szemerédi conjecture for integer sets proved by Bourgain and Chang in 2004. For any $\gamma >0$ there exists $\Lambda (\gamma )>0$ such that for an arbitrary $A\subset \mathbb{N}$, if $|AA|\le K|A|$ then $$E_{+}(A)\le K^{\Lambda }|A{|}^{2+\gamma }.$$