Roth's theorem in the Piatetski-Shapiro primes

• Mariusz Mirek ^[1]
  1. [1] Universität Bonn
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 2, 2015, págs. 617-656
• Idioma: inglés
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• Resumen

  • Let P denote the set of prime numbers and, for an appropriate function hh, define a set Ph={p∈P:∃n∈N p=⌊h(n)⌋}. The aim of this paper is to show that every subset of Ph 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=⌊n1/γ⌋} has this feature. We show this by proving the counterpart of the Bourgain–Green restriction theorem for the set Ph.