Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

Let $t \in \mathbb{N}$, $η>0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-η}$, $q$ not a multiple of the conductor of the exceptional character $χ^*$ (if it exists). Suppose further that, $\max \{p : p | q \} < \exp (\frac{\log x}{C \log \log x}) \; \; {and} \; \; \prod_{p | q} p < x^δ,$ where $C$ and $δ$ are suitable positive constants depending on $t$ and $η$. Let $a \in \mathbb{Z}$, $(a,q)=1$ and $\mathcal{A} = \{n \in (x/2, x]: n \equiv a \pmod{q} \} .$ We prove that there are primes $p_1 < p_2 < ... < p_t$ in $\mathcal{A}$ with $p_t - p_1 \ll qt \exp (\frac{40 t}{9-20 θ}) .$ Here $θ= (\log q) / \log x$.

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