Uniform Diophantine Approximation on the Hecke Group H4

Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation with a given bound. We study uniform Diophantine approximation properties on the Hecke group $\mathbf H_{4}$. For a given real number $\alpha $, we characterize the sequence of $\mathbf H_{4}$-best approximations of $\alpha $ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $\alpha $. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants.