Uniform Diophantine approximation on the Hecke group ${\mathbf{H}}_4$

Abstract

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 $\hat{I}\pm$, we characterize the sequence of ${\mathbf{H}}_4$-best approximations of $\hat{I}\pm$ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $\hat{I}\pm$. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants.

22 pages, 6 figures