Prime-less intervals $[n,\lfloor q\cdot n\rfloor]$ for $q\in \mathbb{Q}, q>1$

Question

Is there $q\in\mathbb{Q}$ with $q>1$ and the following property?

There are infinitely many $n\in\mathbb{N}$ such that there are no primes in $[n,\lfloor q\cdot n\rfloor]$.

Answer 1

No, it follows from the prime number theorem that, for every $\varepsilon>0$ there is $n_0$ such that for every $n>n_0$ there is a prime in the interval $[n,n+\varepsilon n]$. In fact the number of such primes is asymptotically $\varepsilon n/\log n$ as $n\to\infty$.

Of course much better results are known. For example, Baker-Harman-Pintz (2001) showed that for every $n>n_0$ there is a prime in the interval $[n,n+n^{0.525}]$. Moreover, Carneiro-Milinovich-Soundararajan (2019) proved under the Riemann Hypothesis that for every $n>n_0$ there is a prime in the interval $[n,n+(22/25)\sqrt{n}\log n]$.