Dirichlet's approximation theorem says that for every real $\alpha$ and every positive integer $N$, there exist integers $p,q$ with $1 \leq q \leq N$ such that $$ |q\alpha - p| < \frac{1}{N}. $$ It follows that for every real $\alpha$, there are infinitely many integers $p,q$ such that $$ |q\alpha - p| < \frac{1}{|q|} $$
The Thue-Siegel-Roth theorem says that for every irrational algebraic $\alpha$ and every $\epsilon > 0$ there are only finitely many integers $p,q$ such that $$ |q\alpha - p| < \frac{1}{|q|^{1+\epsilon}}. $$
Is it true that for every irrational algebraic $\alpha$ and positive integer $N$ there are only finitely many integers $p,q$ such that $$ |q\alpha - p| < \frac{1}{N}? $$
Let $\alpha$, $N$ be given. By Dirichlet's approximation theorem, there exist infinitely many integers $p,q$ such that $$ |q\alpha - p| < \frac{1}{|q|}. $$ Therefore, there exist infinitely many integers $p,q$ with $|q| > N$ such that $$ |q\alpha - p| < \frac{1}{|q|} < \frac{1}{N}. $$