I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that $$ \left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2} $$ for infinitely many choices of $p$ and $q$. Here $c>0$ is some constant that depends on $\xi$.

My question is: for what (irrational) numbers can we have a precise asymptotic estimate, namely $$ \frac{c_1}{q^2} \leq \left| \xi - \frac{p}{q} \right| \leq \frac{c_2}{q^2} $$ as $q \to +\infty$?