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arXiv · 2503.18517

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

Ayreena Bakhtawar·Dong Han Kim·Seul Bee Lee
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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 $\alpha$, 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.

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Ayreena Bakhtawar, Dong Han Kim, Seul Bee Lee. 2025-03-24. Uniform Diophantine approximation on the Hecke group $\mathbf H_4$. https://arxiv.org/abs/2503.18517

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