Arithmetic Combinatorics

In Michaelmas 2007 Tim Gowers lectured a Part III course on Arithmetic Combinatorics. The course covered Roth's Theorem, Weyl's inequality, quadratic uniformity, quadratic non-uniformity, Bohr neighbourhoods, and Szemerédi's Theorem for progressions of length four. Notes from the course are available here. [These notes are unofficial, and the responsibility for any errors that remain rests firmly with me. Comments and corrections would be gratefully received.]

I wrote an essay on the Lagrangians of Hypergraphs for Part III in Easter 2007, based largely on the theorem of Frankl and Rödl that ‘hypergraphs do not jump’. Although not (at all) related to arithmetic combinatorics, the essay is available here for anyone who is interested.