2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66625We 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.23 pages. Updated references and made some minor changes recommended by the referee. To appear in Annals of MathematicsNumber TheoryClassical Analysis and ODEs11B25; 11P55; 42B15text