Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a result on arithmetic progressions represented by a positive binary form. So my question is whether that is the case, do we already know that a positive binary form represents arbitrarily long arithmetic progressions? These would be primes in this general setting, thus quite different from consecutive integers of course.
My main reference is
David A. Cox,
Primes of the form $x^2 + n y^2.$ He defines the Dirichlet density on page 169. Then he states the
Cebotarev Density Theorem (8.17) on page 170. Finally he gives the Dirichlet density of primes represented by a positive binary form on page 188, Theorem 9.12.
From an earlier answer by David Hansen it would appear that the only missing ingredient is a comparison between Dirichlet density and "relative density." I have never been sure on this point, does a positive binary form represent the same relative density of primes as its Dirichlet density? Anyway, see: Is the Green-Tao theorem true for primes within a given arithmetic progression?
My earlier question, about which I should say that I have come to believe there is no upper bound on the length of intervals represented, despite the great difficulty finding examples: Can a positive binary quadratic form represent 14 consecutive numbers?
On the Green-Tao theorem itself:
http://arxiv.org/abs/math.NT/0404188
http://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem
same-day EDIT: I looked up Dirichlet density on wikipedia. As relates to the earlier David Hansen answer I liked, wikipedia stops short of saying that the Dirichlet density of primes in an arithmetic progression is identical to the relative density. Personally, I cannot see how the relative density could be anything else, but that is just my opinion. So I think I am also asking for references that prove the relative density is equal to the Dirichlet density in some naturally-occurring situations.