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On Systems of Complexity One in the Primes
Published online by Cambridge University Press: 10 May 2016
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.
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- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 1 , February 2017 , pp. 133 - 163
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- Copyright © Edinburgh Mathematical Society 2016
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