TITLE: Estimate number of solutions in the Roth's theorem
QUESTION [10 upvotes]: 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 $\epsilon=1$.
$$\displaystyle \left \lvert \sqrt[3]{2}- \frac{p}{q} \right \rvert < \frac{1}{q^{3}}$$

REPLY [11 votes]: I believe that there is a completely explicit upper bound for $N_\alpha(\epsilon)$ (more generally counting in relative number fields and using more then one, possibly non-archimedean, absolute value) in the following paper:
Robert Gross, 
A note on Roth's theorem.
J. Number Theory 36 (1990), no. 1, 127–132. MR1068678
Of course, it's not going to be pretty.