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 small the smallest large such $q$ can be? Thanks.

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

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

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