Roth's theorem in the primes

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes.Comment: 23 pages. Updated references and made some minor changes recommended by the referee. To appear in Annals of Mathematic

The Green-Tao theorem: an exposition

The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.Comment: 26 pages, 4 figure

The structure theory of set addition revisited

In this article we survey some of the recent developments in the structure theory of set addition.Comment: 38p