On a quantitative refinement of the Lagrange spectrum (Q2781452)

In the late 19th century, Markoff considered the quality of approximation to real numbers by rationals. In particular, he showed that any irrational number \(\alpha\) admits \textit{infinitely many} rationals \(p/q\) such that \(q^2 |\alpha - p/q|\leq 1/\sqrt{5}\). He showed that the golden number required the constant \(\sqrt{5}\). Furthermore, any real equivalent to the golden number also required the same constant. The equivalence on real numbers here is that of eventually agreeing continued fraction expansions. He then found a sequence of natural numbers (the Markoff numbers), \(m_s\), and a corresponding sequence of real numbers, \(\alpha_{m_s}\), such that for any natural number \(r\) and any irrational real \(\alpha\) inequivalent to \(\alpha_{m_s}\) for \(s<r\), there are infinitely many rational numbers \(p/q\) such that \(q^2|\alpha -p/q|\leq 1/\mu_r\). NEWLINENEWLINENEWLINEFor natural numbers \(r\) and \(n\), the authors determine best possible \(C_r(n)\) such that for any irrational real \(\alpha\) inequivalent to \(\alpha_{m_s}\) for \(s<r\), there are at least \(n\) rational numbers \(p/q\) such that \(q^2|\alpha -p/q|\leq 1/C_r(n)\). NEWLINENEWLINENEWLINEThis beautiful result is proven by careful analysis of the partial quotients of the \(\alpha_{m_r}\).