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.
|Translated title of the contribution||Roth's theorem in the primes|
|Pages (from-to)||1609 - 1636|
|Journal||Annals of Mathematics|
|Publication status||Published - May 2005|
Bibliographical notePublisher: Ann Mathematics
Other identifier: IDS Number: 980YR