On systems of complexity one in the primes

2014/03/27 by Kevin Henriot, Henriot, Kevin
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1403.7040

33 pages

arxiv created 2014/05/16 · arxiv updated 2014/05/19

Abstract

Consider a translation-invariant system of linear equations V x = 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(loglog N)^-1/25t, there exists a solution x ∈ A^t to V x = 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 systems of equations of finite complexity by the work of Green and Tao.

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