Edit added for the Bounty: the following question is an old one (without a definitive answer). I faced the same issue, so I hope that someone could complete the given answer
I'm trying to read the book Lecture Notes on Diophantine Analysis by Zannier and he says that the following theorems are equivalent:
Theorem 1. Let $\xi$ be an algebraic real number of degree $d\geq 3$. For every $\epsilon >0$ there is a number $\gamma >0$ such that for all $p,q\in\mathbb{Z}$ $q>0$ $$|\xi-\frac{p}{q}|>\frac{\gamma}{q^{1+\frac{d}{2}+\epsilon}}$$
Theorem 2. Let $\xi$ be an algebraic real number of degree $d\geq 3$. For every $\epsilon >0$ there are only finitely many rationals $\frac{p}{q}$ with $q>0$ such that $$|\xi-\frac{p}{q}|<\frac{1}{q^{1+\frac{d}{2}+\epsilon}}$$
I have already proved that thm 2 implies thm 1, but I'm stuck with thm 1 impies thm 2, Can anybody help me? Thank you
Hint : Suppose, there are infinite many rational numbers $\frac{p}{q}$, such that the condition in $2$ holds for given $d$ and $\epsilon$. Then, there are fractions $\frac{p}{q}$ with arbitary large $q$ under these fractions.