# Additive Combinatorics Lecture Notes

I recently started teaching an “Additive combinatorics” class, and am writing lecture notes for it. So far I put the first three chapters online. I’d appreciate any comments about these, from pointing out serious mistakes, to pointing out minor typos, or even a recommendation for the final topic (which I have not chosen yet). The chapters that are already up are:

• In Chapter 1 we start to study the principle that sets with small doubling must have structure. We prove some basic results such as Ruzsa’s triangle inequality, Plünnecke’s inequality, and variants of Freiman’s theorem.
• Chapter 2 studies the sum-product problem over the reals. In addition to showing the proofs of Elekes and Solymosi, we see how the same techniques can be applied to several other problems.
• Chapter 3 discusses the The Balog-Szemerédi-Gowers theorem. Specifically, we present the variant of Schoen and the variant by Sudakov, Szemerédi, and Vu.
As the quarter progresses, I will keep uploading more chapters.