Revista Matemática Iberoamericana

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Volume 31, Issue 2, 2015, pp. 617–656
DOI: 10.4171/RMI/848

Published online: 2015-07-16

Roth's theorem in the Piatetski-Shapiro primes

Mariusz Mirek[1]

(1) Universität Bonn, Germany

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 $\mathbf{P}_{h}$.

Keywords: Arithmetic progressions, trigonometric sums, Hardy–Littlewood majorant problem, discrete restriction

Mirek Mariusz: Roth's theorem in the Piatetski-Shapiro primes. Rev. Mat. Iberoamericana 31 (2015), 617-656. doi: 10.4171/RMI/848