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}\colon \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} .

Locations

• Revista Matemática Iberoamericana - View - PDF
• arXiv (Cornell University) - View - PDF