Approximation results for $\alpha$-Rosen fractions

In this article we generalize Borel's classical approximation results for the regular continued fraction expansion to the $\alpha$-Rosen fraction expansion, using a geometric method. We use $\alpha$-Rosen fractions to give a Haas-Series-typeresult about all possible good approximations for the $\alpha$ for which the Legendre constant is larger than the Hurwitz constant.