# Roth's theorem in the Piatetski-Shapiro primes

### Mariusz Mirek

Universität Bonn, Germany

## Abstract

Let $P$ denote the set of prime numbers and, for an appropriate function $h$, define a set $P_{h}={p∈P:∃_{n∈N}p=⌊h(n)⌋}$. The aim of this paper is to show that every subset of $P_{h}$ 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., ${p∈P:∃_{n∈N}p=⌊n_{1/γ}⌋}$ has this feature. We show this by proving the counterpart of the Bourgain–Green restriction theorem for the set $P_{h}$.

## Cite this article

Mariusz Mirek, Roth's theorem in the Piatetski-Shapiro primes. Rev. Mat. Iberoam. 31 (2015), no. 2, pp. 617–656

DOI 10.4171/RMI/848