Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

Let $t\in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q\le x^{5\mathrm{/}12-\eta }$, $q$ not a multiple of the conductor of the exceptional character ${\chi }^{\ast }$ (if it exists). Suppose further that, $$\max \{p:p\mathrm{\mid }q\}<\exp (\frac{\log x}{C\log \log x})and\prod _{p\mathrm{\mid }q}p<x^{\delta },$$ where $C$ and $\delta$ are suitable positive constants depending on $t$ and $\eta$. Let $a \in $\mathbb{Z}$$,$(a,q)=1$ and $$\mathcal{A}=\{n\in (x\mathrm{/}2,x]:n\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}q)\}\mathrm{.}$$ We prove that there are primes $p_1<p_2<\mathrm{.}\mathrm{.}\mathrm{.}<p_t$ in $\mathcal{A}$ with $$p_t-p_1\ll qt\exp (\frac{40t}{9-20\theta })\mathrm{.}$$ Here $\theta =(\log q)\mathrm{/}\log x$.