JournalsrmiVol. 31, No. 2pp. 617–656

Roth's theorem in the Piatetski-Shapiro primes

  • Mariusz Mirek

    Universität Bonn, Germany
Roth's theorem in the Piatetski-Shapiro primes cover
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Abstract

Let P\mathbf{P} denote the set of prime numbers and, for an appropriate function hh, define a set Ph={pP ⁣:nN p=h(n)}\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 Ph\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<γ<171/72<\gamma<1, i.e., {pP:nN p=n1/γ}\{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 Ph\mathbf{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