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 the Bourgain–Green restriction theorem for the set ${\mathbf{P}}_h$.