TITLE: The difference between consecutive primes in arithmetic progressions
QUESTION [9 upvotes]: Let $\pi(x)=\sum_{p\leq x}$ denote the prime counting function. A well known result of Baker, Harman, and Pintz on prime gaps states that for $x\geq y\geq x^{0.525}$ we have that
$$\pi(x+y)-\pi(x)\gg \frac{y}{\log x}.$$
Let $\pi(x;q,a)$ denote the number of primes $p\leq x$ such that $p\equiv a\pmod{q}$. Is there an analogous version of this theorem for $\pi(x;q,a)$?

Question: What is the smallest growing function $f(x,q)$ such that for any $x\geq y\geq f(x,q)$ we have that $$\pi(x+y;q,a)-\pi(x;q,a)\gg \frac{y}{\log x}?$$

Corollary 13.8 of Montgomery and Vaughn's Multiplicative Number Theory states that under Generalized Riemann Hypothesis $$\psi(x;q,a)=\sum_{\begin{array}{c}
n\leq x\\
n\equiv a\ (q)
\end{array}}\Lambda(n)=\frac{x}{\phi(q)}+O\left(x^{1/2}\log^{2}x\right).$$
From this Corollary, it follows that under GRH the desired lower bound holds for $$f(x,q) =  \phi(q)x^\frac{1}{2}\log^{2}(x).$$
I'm aware of a 1933 result of Heilbronn (MR1545353), which I believe (my German isn't great) proves that for fixed $q,a$, there exists $\delta>0$ such that for $x\geq y\geq x^{1-\delta}$ $$\pi(x+y;q,a)-\pi(x;q,a)\sim \frac{y}{\phi(q)\log x}.$$
Are there any more recent results on this topic?

REPLY [4 votes]: Unconditionally, one can take $f(x,q)=x^{0.525}$, provided that $q$ is in the Siegel-Walfisz range $q\leq (\log x)^A$.  There exists a (suitably small) constant $\delta>0$ such that one can take $f(x,q)=x^{0.525}$ for all except a density zero subset of $q\leq x^{\delta}$.
This follows from a certain Bombieri-Vinogradov-type result for a carefully constructed lower bound for the indicator function of the primes (due to  Kumchev, but with $0.525$ replaced by $0.53$) combined with the second moment bound for $L(s,\chi)^2$ times a Dirichlet polynomial (due to Harman, Watt, and Wong).  Full details on how to combine these two results can be found in the work of Alweiss and Luo (see Theorem 2.3 therein).  One then takes $Q = (\log x)^A$.