Roth's theorem in the Piatetski-Shapiro primes

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Abstract

Let P denote the set of prime numbers and, for an appropriate function h, 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.

Original languageEnglish (US)
Pages (from-to)617-656
Number of pages40
JournalRevista Matematica Iberoamericana
Volume31
Issue number2
DOIs
StatePublished - 2015

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Arithmetic progressions
  • Discrete restriction
  • Hardy-Littlewood majorant problem
  • Trigonometric sums

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