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 $71/72<γ<1$, i.e. $\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor n^{1/γ}\rfloor\}$ has this feature. We show this by proving the counterpart of Bourgain--Green's restriction theorem for the set $\mathbf{P}_{h}$.

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