Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-08T08:33:15.195Z Has data issue: false hasContentIssue false
  • English
  • Français

On Systems of Complexity One in the Primes

Part of: Diophantine equations Sequences and sets

Published online by Cambridge University Press:  10 May 2016

Kevin Henriot
Kevin Henriot*
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, CP 6128, Succursale Centre-Ville, Montréal, Québec H3C 3J7, Canada (henriot@dms.umontreal.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a translation-invariant system of linear equations Vx = 0 of complexity one, where V is an integer r × t matrix. We show that if A is a subset of the primes up to N of density at least C(log logN)–1/25t, there exists a solution x ∈ At to Vx = 0 with distinct coordinates. This extends a quantitative result of Helfgott and de Roton for three-term arithmetic progressions, while the qualitative result is known to hold for all translation-invariant systems of finite complexity by the work of Green and Tao.

Keywords

linear equations in primesRoth theoremadditive cominatoricspseudo-randomnesscomplexity of linear systems

MSC classification

Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 11D04: Linear equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 1 , February 2017 , pp. 133 - 163
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)