On a theorem of Borel on Diophantine approximation (Q6624953)

In 1903, Borel proved that for every real number \(\alpha\) and every three consecutive convergents of its continued fraction, there is at least one of them, say \(\frac{p}{q}\), which satisfies \N\[\biggl|\alpha-\frac{p}{q}\biggr|<\frac{1}{Cq^2}\] \Nfor a constant \(C=\sqrt{5}\).\N\NIn the present paper the authors improve the result, replacing the constant \(C\) by a function \(D(q)=\sqrt{5}+\frac{4-5\sqrt{5}+\sqrt{61}}{2q^2}\) which is greater than \(C\):\N\NFor every real number \(\alpha\) and every three consecutive convergents of its continued fraction, there is at least one of them, say \(\frac{p}{q}\), which satisfies \N\[\biggl|\alpha-\frac{p}{q}\biggr|<\frac{1}{D(q)q^2} .\]\N\NThe authors also show that this function \(D(q)\) is the best possible. The boundary cases are the numbers \(\alpha=\frac{1}{2+\frac{8}{3\sqrt{5}+\sqrt{61}}}\) and \(1-\alpha\).