A transference principle for systems of linear equations, and applications to almost twin primes

The transference principle of Green and Tao enabled various authors to transfer Szemer\'edi's theorem on long arithmetic progressions in dense sets sparse integers, mostly primes. In this paper, we provide a which applies general affine-linear configurations finite complexity. We illustrate the broad applicability our with case almost twin primes, by mean either Chen primes or "bounded gap primes", as well form $x^2+y^2+1$. Thus, show that these existence solutions complexity systems linear equations is determined natural local conditions. These applications rely recent work last two Bombieri-Vinogradov type estimates for nilsequences.