Bourgain-Chang's proof of the weak Erd\H{o}s-Szemer\'edi conjecture

TL;DR

This paper explains Bourgain and Chang's proof of a weak form of the Erdős-Szemerédi conjecture, establishing bounds on additive energy for sets with small product sets.

Contribution

It provides an exposition of Bourgain and Chang's 2004 proof of a weak Erdős-Szemerédi conjecture for integer sets.

Findings

Bound on additive energy in terms of product set size

Existence of a function b3 b5 0 such that the energy bound holds

Proof techniques from Bourgain and Chang's original work

Abstract

This is an exposition of the following `weak' Erd\H{o}s-Szemer\'edi 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 $|AA| \leq K|A|$ then $$E_{+}(A) \leq K^{\Lambda}|A|^{2+\gamma}.$$