Complex Diophantine Approximations and Cusp Excursions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913174122397696 |
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| author | Dalaklis, Nathan He, Yan Mary |
| author_facet | Dalaklis, Nathan He, Yan Mary |
| contents | We study the Hausdorff dimension spectrum of asymptotic approximation rates of complex Diophantine approximation and that of the asymptotic average excursion time of cusp excursions on the Bianchi orbifold $\mathbb{H}^3/\operatorname{PSL}(2,\mathbb {Z}[i])$ via a unified approach using the Hurwitz map. In particular, we construct a conformal graph directed system (CGDS) for the Hurwitz map and show that the Lyapunov exponent of the Hurwitz CGDS simultaneously captures the asymptotic approximation rate and the the asymptotic average excursion time. Applying the multifractal analysis of Lyapunov exponents for this system, we obtain a formula and real-analyticity for the Hausdorff dimension spectrum functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_31502 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complex Diophantine Approximations and Cusp Excursions Dalaklis, Nathan He, Yan Mary Dynamical Systems 37D40, 11J70 (Primary) 37D35, 28A80, 37A05 (Secondary) We study the Hausdorff dimension spectrum of asymptotic approximation rates of complex Diophantine approximation and that of the asymptotic average excursion time of cusp excursions on the Bianchi orbifold $\mathbb{H}^3/\operatorname{PSL}(2,\mathbb {Z}[i])$ via a unified approach using the Hurwitz map. In particular, we construct a conformal graph directed system (CGDS) for the Hurwitz map and show that the Lyapunov exponent of the Hurwitz CGDS simultaneously captures the asymptotic approximation rate and the the asymptotic average excursion time. Applying the multifractal analysis of Lyapunov exponents for this system, we obtain a formula and real-analyticity for the Hausdorff dimension spectrum functions. |
| title | Complex Diophantine Approximations and Cusp Excursions |
| topic | Dynamical Systems 37D40, 11J70 (Primary) 37D35, 28A80, 37A05 (Secondary) |
| url | https://arxiv.org/abs/2605.31502 |