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On Systems of Complexity One in the Primes

Part of: Sequences and sets Diophantine equations

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 ([email protected])
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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 

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