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 H₄. For a given real number α, we characterize the sequence of H₄-best approximations of α and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of α. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants.