Sharpening Vahlen’s result in Diophantine approximation

In this paper we refine Vahlen’s 1895 result in Diophantine approximation by providing sharper bounds for the approximation coefficients. In particular, when at least one of the partial quotients an or an+1 of the regular continued fraction expansion [a0;a1,a2,…] of x is 1. Improvements of Vahlen’s result were given by Jaroslav Hančl (2015), Hančl and Silvie Bahnerová (2021), and Dinesh Sharma Bhattarai (2023). The approach of the present paper is very different from that of Hančl {et al}. The geometrical methods used in this paper not only offer a significant improvement over Vahlen’s result, but also yield new insights that can contribute to improving Borel’s classical constant.