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On Systems of Complexity One in the Primes

Published online by Cambridge University Press:  10 May 2016

Kevin Henriot*
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, CP 6128, Succursale Centre-Ville, Montréal, Québec H3C 3J7, Canada ([email protected])

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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