Additive Combinatorics

Fridays 2-3.30pm in 2.135, starting September 30th. As I noted in the lectures, there is *tea* at 3.30 in the common room, which is room 2.290. Discussions on additive combinatorics, or anything else of interest, would be welcome there.

Course description.

• 30/9/05 Lecture 1: Plunnecke-Ruzsa inequalities.
• 7/10/05 Lecture 2: Discrete Fourier. Bogolyubov. Chang's Theorem
• 14/10/05 Lecture 3: Chang-Bogolyubov. Minkowski's second theorem. Progressions in Bohr sets
• 21/10/05 Lecture 4: Proof of Freiman's Theorem, sketch of refinements
• 28/10/05 No lecture (I'm away in Bristol)
• 4/11/05 Lecture 5: We started working on Bourgain-Katz-Tao/Bourgain-Konyagin. Here are some notes.
• 11/11/05 No lecture (Veterans' Day)
• 18/11/05 Lecture 6: Conclusion of the proof of BKT/BK.
• 25/11/05 Yet another day off. I thought Americans liked hard work??
• 2/12/05 Lecture 7: Gowers' proof of Szemeredi's theorem for 4-term APs,I
• 9/12/05 Lecture 8: Gowers' proof of Szemeredi's theorem for 4-term APs, II

Materials:

• Some old notes of mine from a conference in Edinburgh in 2002 formed the basis for the first 3 and a bit lectures. I'm going to update these soon and combine them with some new stuff (for example the notes below).
• Here are some recent notes I wrote, explaining the last part of Mei-Chu Chang's paper in which she obtains very good bounds on the dimension in Freiman's theorem.
• The webpage of Terry Tao is very recommended. You'll find notes from a course he gave in 2003, and sample chapters from his forthcoming book with Van Vu, which will become the standard reference in this subject.
• The superb Palo Alto open problems, collected by Ernie Croot and Seva Lev.
• Tim Gowers let me put up his wonderful example sheets from a course given in Cambridge in 1999. Here they are: Sheet 1 Sheet 2 Sheet 3. Warning: not all of the techniques you need to do these questions are in my course. Also (a Cambridge tradition) some of these questions are *hard*, even the Q1s).