Roth's theorem in the Piatetski-Shapiro primes

Abstract

Let P denote the set of prime numbers and, for an appropriate function hh, define a set Ph={p∈P:∃n∈N p=⌊h(n)⌋}. The aim of this paper is to show that every subset of Ph 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∈P:∃n∈N p=⌊n1/γ⌋} has this feature. We show this by proving the counterpart of the Bourgain–Green restriction theorem for the set Ph.