Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

Abstract: Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists). Suppose further that, [ \max {p : p | q } < \exp (\frac{\log x}{C \log \log x}) \; \; {and} \; \; \prod_{p | 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/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 \theta}) . ] Here $\theta = (\log q) / \log x$.