On systems of complexity one in the primes

Abstract

Consider a translation-invariant system of linear equations $Vx=0$ of complexity one, where $V$ is an integer $r\times t$ matrix. We show that if $A$ is a subset of the primes up to $N$ of density at least $C(\log \log N{)}^{-1/25t}$, there exists a solution $x\in A^t$ 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 systems of equations of finite complexity by the work of Green and Tao.

arXiv comment33 pages