I am interested whether there are functions $\hat{b}(q)$ and $\check{b}(q)$ such that for any $\alpha\in [0,1)\setminus \mathbb{Q}$ we have for only finitely many $q's$
$$ \check{b}(q) \leq \Big\vert \alpha-\frac{p}{q} \Big\vert \quad \text{and} \quad \Big\vert \alpha-\frac{p}{q} \Big\vert \leq \hat{b}(q) \quad \text{for some}\;\; p\in \mathbb{Z}. $$
In a sense, I am wondering about the possible best case approximation and worst case approximation of irrational numbers. Looking at the Diophantine approximation entry on Wikipedia, it seems as though if such functions exist then
$$ \check{b}(q)=O\Big( \frac{1}{q} \Big) \quad \text{and} \quad \hat{b}(q)=o\Big(\frac{1}{q^n}\Big) \;\; \text{for all} \;\; n\in \mathbb{N}. $$
Are there more explicit asymptotic bounds ? Can we say something like $\hat{b}(q)=o(e^{-q})$ and $\frac{\check{b}(q)}{q^{1+\epsilon}} \overset{q\to \infty}{\to} \infty$ for every $\epsilon>0$?
I assume something like this is known and I have not found this in a short search online. I am unused to these notions so I apologize if this is a silly question.