Additive Combinatorics Lecture Notes (part 2)

I finished my additive combinatorics class, and placed all of the lecture notes in the pdf files page. This quarter was rather short and I did not get to do several topics I had in mind. Perhaps I’ll add notes for some of these at some point. For now, let me just list the chapters that did not appear in the previous post.

  • Chapter 4 is about arithmetic progressions in dense sets. It includes Behrend’s construction, Meshulam’s theorem, and Roth’s theorem. Due to the recent developments, Meshulam’s theorem already seems somewhat outdated…
  • Chapter 5 consists of Sander’s proof of the quasi-polynomial Freiman-Ruzsa theorem in {\mathbb F}_2^n  (following Lovett’s presentation). This includes proving a special case of the probabilistic technique of Croot and Sisask.
  • Chapter 6 presents the technique of relying on the third moment energy. It then uses this technique to study convex sets. I hope to also add a variant of the Balog-Szemerédi-Gowers theorem.