On Systems of Complexity One in the Primes
2016 ◽
Vol 60
(1)
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pp. 133-163
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Keyword(s):
AbstractConsider a translation-invariant system of linear equationsVx= 0 of complexity one, whereVis an integerr×tmatrix. We show that ifAis a subset of the primes up toNof density at leastC(log logN)–1/25t, there exists a solutionx∈ AttoVx= 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.
1998 ◽
Vol 90
(5)
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pp. 2398-2403
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