Roth's Theorem in the Piatetski-Shapiro primes

Abstract: 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 $\mathbf{P}_{h}$.