On a Theorem of Nathanson on Diophantine Approximation

In 1974, M. B. Nathanson proved that every irrational number $\alpha$ represented by a simple continued fraction with infinitely many elements greater than or equal to $k$ is approximable by an infinite number of rational numbers $p\mathrm{/}q$ satisfying $\mathrm{\mid }\alpha -p\mathrm{/}q\mathrm{\mid }<1\mathrm{/}(\sqrt{k^2+4}{q}^2)$. In this paper we refine this result.