Let denote the set of prime numbers and, for an appropriate function , define a set . The aim of this paper is to show that every subset of having positive relative upper density contains a nontrivial three-term arithmetic progression. In particular the set of Piatetski-Shapiro primes of fixed type , i.e., has this feature. We show this by proving the counterpart of the Bourgain–Green restriction theorem for the set .
Cite this article
Mariusz Mirek, Roth's theorem in the Piatetski-Shapiro primes. Rev. Mat. Iberoam. 31 (2015), no. 2, pp. 617–656DOI 10.4171/RMI/848