It is known that there exists a prime between $n$ and $n+n^{\epsilon}$ for $n$ large enough, where the best known $\epsilon$ is 0.535. Therefore, given a large enough prime $q$, choose a prime $p$ between $q\alpha$ and $q\alpha+(q\alpha)^{\epsilon}$ to get $$|\alpha-p/q|<Cq^{\epsilon-1},$$ where $C$ depends only on $\alpha$. Assuming the Reimann Hypothesis, this bound can be improved to $$|\alpha-p/q|<Cq^{-1/2}\log q,$$ for infinitely many pairs of primes $(p,q)$. With recent results on prime gaps, this bound is extremely better when $\alpha=1$. One can then ask this question:

${\bf Question:}$ Let $\alpha$ be a positive real number. Does there always exist infinitely many pairs of primes $(p,q)$ such that $$|\alpha-p/q|<C/q,$$ where $C$ is a universal constant?

It is known that there exists a prime between $n$ and $n+n^{\epsilon}$ for $n$ large enough, where the best known $\epsilon$ is 0.525. Therefore, given a large enough prime $q$, choose a prime $p$ between $q\alpha$ and $q\alpha+(q\alpha)^{\epsilon}$ to get $$|\alpha-p/q|<Cq^{\epsilon-1},$$ where $C$ depends only on $\alpha$. Assuming the Reimann Hypothesis, this bound can be improved to $$|\alpha-p/q|<Cq^{-1/2}\log q,$$ for infinitely many pairs of primes $(p,q)$. With recent results on prime gaps, this bound is extremely better when $\alpha=1$. One can then ask this question:

${\bf Question:}$ Let $\alpha$ be a positive real number. Does there always exist infinitely many pairs of primes $(p,q)$ such that $$|\alpha-p/q|<C/q,$$ where $C$ is a universal constant?