On systems of complexity one in the primes

Consider a translation-invariant system of linear equations $V x = 0$ 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 $V x = 0$ 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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