Uniform Diophantine approximation on the Hecke group $\mathbf H_4$

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.