Let $\mathbf{ P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\mathbf{ P}_{ h}=\{ p\in\mathbf{ P}: \exists_{ n\in\mathbb{ N}}\ p=\lfloor h(n)\rfloor\}$. The aim of this paper is to show that every subset of $\mathbf{ P}_{ h}$ having positive relative upper density contains a nontrivial three-term arithmetic progression. In particular the set of Piatetski--Shapiro primes of fixed type $71/72<\gamma<1$, i.e. $\{ p\in\mathbf{ P}: \exists_{ n\in\mathbb{ N}}\ p=\lfloor n^{ 1/\gamma}\rfloor\}$ has this feature. We show this by proving the counterpart of Bourgain--Green's restriction theorem for the set $\mathbf{ P}_{ h}$.
Categories: math.CA