Roth's Theorem in the Piatetski-Shapiro primes

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 $71\mathrm{/}72<\gamma <1$, i.e. $\{p\in \mathbf{P}:{\mathrm{\exists }}_{n\in \mathbb{N}}p=\lfloor n^{1\mathrm{/}\gamma }\rfloor \}$ has this feature. We show this by proving the counterpart of Bourgain–Green’s restriction theorem for the set ${\mathbf{P}}_h$.