I originally planned for Chapter 9 of the book to be about additive combinatorics. However in the interests of completing the book before I die of old age, I am planning to make some cuts. It now looks like Chapter 9 (being a rather standalone topic in the context of the book) will hit the cutting room floor. Drafts of some parts of it are already written, and I decided to publish here a “deleted scene” — the Quasipolynomial Freiman–Ruzsa Theorem highlight. It seemed timely to me in light of several other very nice surveys/expositions/extensions that have appeared recently. Shachar Lovett has posted one survey here, and Eli Ben-Sasson, Noga Ron-Zewi, Madhur Tulsiani, and Julia Wolf have just posted a paper on the topic. Also, Sanders himself has a 1.5-page proof of his result in ${\mathbb F}_2^n$ in a manuscript called On the Bogolyubov–Ruzsa Lemma in ${\mathbb F}_2^n$; I don’t know if it’s appeared anywhere, but I’m sure he’ll send you a copy if you ask nicely 
Below is the draft of what was to appear in the book on this topic. I would like to emphasize that its interpretation/proof of the Croot–Sisask result is based on joint discussions with Eric Blais and Shachar Lovett.
Continue reading Additive Combinatorics and the Polynomial Freiman–Rusza Conjecture
Recent comments