Roth's Theorem in the Piatetski-Shapiro primes

TL;DR

This paper proves that subsets of Piatetski-Shapiro primes with positive density contain 3-term arithmetic progressions, extending Roth's theorem to these sparse prime subsets using a restriction theorem approach.

Contribution

It establishes Roth's theorem for Piatetski-Shapiro primes of certain types by adapting restriction theorems to these sets, a novel extension in additive number theory.

Findings

Subsets of Piatetski-Shapiro primes with positive density contain 3-term arithmetic progressions.

Proves a restriction theorem analogue for Piatetski-Shapiro primes.

Extends classical results on primes to a new class of sparse prime sets.

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 Bourgain--Green's restriction theorem for the set $\mathbf{P}_{h}$.