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.

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 nontrivial effective result can be proved. Most notable is the BakerFeldman theorem which provides such a result with an exponent of the form $\deg \alpha  \epsilon$ (instead of $2+\epsilon$) for a suitable small positive $\epsilon$. 


Of course, there may be none or there may be a very small one (if $\alpha<1$ then $q=1,h=0$ works). If there is a small very good approximation, the others must be very large. There are lots of papers on effective diophantine approximation dealing with this. I recall there was a paper of Davenport and Roth which first established such a bound. 

