Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

For any $\epsilon>0$ and any non-exceptional modulus $q\ge3$, we prove that, for $x$ large enough ( $x\ge \alpha_{\epsilon}\log^2 q$), the interval $\left[ e^x,e^{x+\epsilon }\right]$ contains a prime $p$ in any of the arithmetic progressions modulo $q$. We apply this result to establish that every integer $n$ larger than $\exp(71\,000)$ is a sum of seven cubes.