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Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.

Now, consider an arithmetic progression with starting term $a$ and common difference $d$. According to Dirichlet's theorem(suitably strengthened), the primes are "equally distributed" in each residue class modulo $d$. Therefore we imagine that the Green-Tao theorem should still be true if instead of primes we consider only those positive primes that are congruent to $a$ modulo $d$. That is, Green-Tao theorem is true for primes within a given arithmetic progression.

Question: Is something known about this stronger statement?

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The Green-Tao is true for any subset of the primes of positive relative density; the primes in a fixed arithmetic progression to modulus $d$ have relative density $1/\phi(d)$.

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    $\begingroup$ can't argue with that.... $\endgroup$ – Ben Green May 20 '10 at 19:45
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    $\begingroup$ Neither can I... $\endgroup$ – Terry Tao May 24 '10 at 6:35
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    $\begingroup$ This is why I love Mathoverflow... $\endgroup$ – Koundinya Vajjha Sep 19 '10 at 13:34
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    $\begingroup$ Well downvote me if you will, but I think the answer should say that the residue mod $d$ must be coprime to $d$, otherwise there is at most one prime in that residue class, so no nontrivial arithmetic progression in primes in that residue class. $\endgroup$ – plm Dec 4 '12 at 1:24
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    $\begingroup$ @KoundinyaVajjha, I think that another reason to love MO is the exactly 256 upvotes (each) that Ben Green and Terry Tao got. $\endgroup$ – LSpice Nov 23 '16 at 17:15

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