Estimate number of solutions in the Roth's theorem 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 $\epsilon=1$.
$$\displaystyle \left \lvert \sqrt[3]{2}- \frac{p}{q} \right \rvert < \frac{1}{q^{3}}$$
 A: For a fixed $\alpha$, the number $N_{\alpha}(\epsilon)$ is bounded by a polynomial function of $1/\epsilon$. The proof of this requires either Faltings's product theorem, or Esnault and Viehweg's multidimensional Dyson lemma. See section 6.5 of Bombieri and Gubler's book (Heights in Diophantine Geometry), with particular attention to point 6.5.8.
Roth's original proof gives a bound exponential in $\epsilon^{-2}$. An explicit form of this bound is worked out in Davenport and Roth's paper Rational approximations to algebraic numbers (Mathematika, 1955), and is also sketched in section 6.5 of Bombieri and Gubler's book. 
You might be more interested in the dependence on the degree $d$ and absolute logarithmic height $h(\alpha)$ of $\alpha$ (I should also add the absolute value $|\alpha|$ for a cleaner dependence), and not so much on $\epsilon$. For that, the bound that comes out of these methods is
$$
\ll_{\epsilon} 1 + \log^+{h(\alpha)} + \log^+{|\alpha|} + (\log{d})^{O(1)}.
$$
A: 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.
