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Let $\mathbb{P}_{a, q}$ denote the set of primes congruent to $a$ modulo $q$. Are there any estimates for the number of $3$-Arithmetic Progressions in the set $\mathbb{P}_{a, q}\cap [1, X]$, where $(a, q) = 1$ and $q$ is large as a function of $X$?

For $q$ at most a constant, such an estimate can be given thanks to Green [1]. One can take a slightly better $q$ thanks to the work of Helfgott and Roton [2]. Both works deal with a much more general setting of an arbitrary dense subset of primes. Can one gain in how large of a modulus $q$ is admissible if one restricts specifically to $\mathbb{P}_{a, q}\cap [1, X]$?

To be more precise, I want to know how large of a modulus $q$ one can take as a function of $X$ such that one has a result of the kind $$|\{(x, y, z): x, y, z\in \mathbb{P}_{a, q}\cap [1, X]\text{ and }x+z=2y\}|\gg \frac{1}{\phi(q)^3}\frac{X^2}{(\log X)^3}.$$

[1] https://arxiv.org/pdf/math/0302311

[2] https://arxiv.org/abs/0912.1842

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    $\begingroup$ please supply details, references, clear expressions where available $\endgroup$
    – kodlu
    Commented Apr 25 at 15:30
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    $\begingroup$ @kodlu I have made some edits, I hope these help improve the quality. $\endgroup$
    – MathPanda
    Commented Apr 25 at 16:27
  • $\begingroup$ @MathPanda Do you want $\frac{1}{\phi(q)^2}$ instead of $\frac{1}{\phi(q)^3}$? $\endgroup$
    – mathworker21
    Commented Apr 25 at 17:37
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    $\begingroup$ Maybe mention that q is a function of X much earlier. $\endgroup$
    – Daniel Asimov
    Commented Apr 26 at 3:18
  • $\begingroup$ @mathworker21 I think the result holds with $1/\phi(q)^2$ as you pointed out, but I am fine with any lower bound that goes to infinity as $X$ goes to infinity. $\endgroup$
    – MathPanda
    Commented Apr 27 at 3:51

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