There is a fundamental theorem in Diophantine approximation :

For all algebraic irrational $\alpha$ $$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \epsilon}}$$ with $\epsilon>0$, has finitely many solutions.

can we estimate number of solutions $N_{\alpha}(\epsilon)$?

for instance :

what is the upper bound of $N_{\sqrt[3]{2}}(1)$?

number of solutions for $\sqrt[3]{2}$, with $\epsilon=1$.

$$\displaystyle \left \lvert \sqrt[3]{2}- \frac{p}{q} \right \rvert < \frac{1}{q^{3}}$$