Quoting from Green-Tao, "Linear equations in primes" (especially Cor. 1.9 in https://arxiv.org/pdf/math/0606088.pdf), any system of linear forms of finite complexity and without any local obstructions will assume simultaneous prime values infinitely often.

(E.g., $(X,Y,X+Y)$ is of finite complexity, but with a local obstruction at 2, since one of these is always even. On the other hand $(X,Y,X+2Y)$ has no local obstruction, so there are infinitely many primes $x,y$ such that $x+2y$ is also prime.)

My question is: Does this result remain true when, instead of asking for prime values, one asks for values in some positive density *subset* of the primes? This kind of strengthening is true e.g. for the famous special case $X, X+Y, X+2Y,...$ of arithmetic progressions in primes, but I couldn't find it in the general case.

Again, to illustrate what I mean: Let's take the set $S$ of all primes $\equiv 1$ mod $4$. Then the system $(X,Y,3X+4Y)$ will be obstructed, since for $x,y\equiv 1$ mod $4$, one has $3x+4y\equiv 3$ mod $4$. On the other hand, $(X,Y,5X+4Y)$ would not be obstructed, so should be expected to have infinitely many values in $S^3$. Is this known in generality?

[I should add that for the last example, one may of course just replace X and Y by 4X+1 and 4Y+1; but of course there are positive density subsets of primes which are not just residue classes. I'm interested in those as well.]

[EDIT: To be more explicit: Let S be a Chebotarev set of primes, i.e. the set of all primes with some given Frobenius in some given Galois extension $K/\mathbb{Q}$. Then there will usually be a congruence condition (but not only!) on $S$ coming from the largest cyclotomic subfield of $K$. So there is some $N$ and some smallest possible finite union $U$ of mod $N$ residue classes containing all of $S$. A local ostruction for a system of linear forms now definitely occurs if there are no integer values simultaneously lying in $U$. I'm asking if that is the only obstruction for values to simultaneously lie in $S$.

Example: if S is the set of primes which decompose completely in the splitting field of $x^3-x^2+1$, then the largest abelian subfield is $\mathbb{Q}(\sqrt{-23})$. To be decomposed in this field means to be a square mod 23 (leaving 11 of 23 possible residue classes). I'm tempted to conjecture that a system of linear forms of finite complexity which has integer values simultaneously lying in one of these 11 classes (i.e. not obstructed at 23), and which for any other prime $p$ has integer solutions simultaneously coprime to $p$ (i.e. not obstructed at $p$) will have prime values simultaneously lying in $S$.]

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