The following question came up in my arithmetic geometry course yesterday. Suppose $\alpha$ is an irrational real algebraic integer, and suppose $\epsilon >0$ is given. Then by Roth's theorem there are at most finitely many rational numbers $\frac{h}{q}$ with $\gcd(h,q)=1$, $q>1$, such that 
$$
\left| \alpha - \frac{h}{q}\right| < \frac{1}{q^{2+\epsilon}}.
$$
Are there any results on how large such $q$ can be? Thanks. 



Question related to Diophantine approximations and Roth's theorem
My other answer was for the first version of this question. The question has now been changed completely. As Antoine mentioned in his comment, an effective Roth's theorem is not known in general. Finding such a result is the probably the main open problem in Diophantine approximation. There some instances in which a non-trivial effective result can be proved. Most notable is the Baker-Feldman theorem which provides such a result with an exponent of the form $\deg \alpha - \epsilon$ (instead of $2+\epsilon$) for a suitable small positive $\epsilon$.


