Second basic theorem of Hurwitz (Q282133)

The author proves for real numbers \(x\) (not equivalent to the golden ratio \(\frac{1\pm\sqrt{5}}{2}\)) that there are infinitely many integer pairs \((p,q)\) such that the Diophantine inequality \[ \left| x-\frac{p}{q}\right|\leq \frac{1}{\frac{q^{2}\sqrt{8}}{2}\big(1+\sqrt{1+\frac{1}{2q^{2}}\big)}} \] holds. This improves a result of Hurwitz where the right hand side is \(\frac{1}{\sqrt{8}}\cdot \frac{1}{q^{2}}\).