Roth's theorem in the Piatetski-Shapiro primes

Let P denote the set of prime numbers and, for an appropriate function h , define a set Pₕ=\{p ∃ₙ\ p= h(n)\} . The aim of this paper is to show that every subset of Pₕ having positive relative upper density contains a nontrivial three-term arithmetic progression. In particular the set of Piatetski-Shapiro primes of fixed type 71/72<γ<1 , i.e., \{p: ∃ₙ\ p= n1/γ\} has this feature. We show this by proving the counterpart of the Bourgain–Green restriction theorem for the set Pₕ .