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This is an old article about Green-Tao's work(transference).

we deduce from Chen's theorem, Roth's theorem, and a transference principle that there are infinitely many arithmetic progressions p1 < p2 < p3 of primes, such that pi + 2 is either a prime or a product of two primes for each i = 1, 2, 3.

Denote the Gowers Inverse conjecture by 'GI(s)' and denote the M¨obius and nilsequences conjecture by 'MN(s)', Our results are therefore unconditional in the case s = 2, and in particular we can obtain the expected asymptotics for the number of 4-term

it is reasonable to conjecture that an analogous result to Theorem 1.3 also holds in higher dimensions.This is however still open even in the linear case, the key difficulty being that the tensor product of pseudorandom measures is not pseudorandom.

The setting of this paper was deliberately placed at a midpoint between graph theory and ergodic theory, and the author hopes that it illuminates the analogies and interconnections between these two subjects.