The Markoff and Lagrange spectra on the Hecke group H4
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912895227396096 |
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| author | Kim, Dong Han Sim, Deokwon |
| author_facet | Kim, Dong Han Sim, Deokwon |
| contents | We consider the Markoff spectrum and the Lagrange spectrum on the Hecke group $\mathbf H_4$. They are identical to the Markoff and Lagrange spectra of the unit circle. The Markoff spectrum on $\mathbf H_4$ is also known as the Markoff spectrum of index 2 sublattices by Vulakh and the Markoff spectrum of 2-minimal forms or $C$-minimal forms by Schmidt. They characterized the spectrum up to the first accumulation point, independently. We show that, after the first accumulation point, both spectra have positive Hausdorff dimension. Then we find gaps in the spectra and give a bound on Hall's ray. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_05441 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Markoff and Lagrange spectra on the Hecke group H4 Kim, Dong Han Sim, Deokwon Number Theory Dynamical Systems 11J06, 11J70 We consider the Markoff spectrum and the Lagrange spectrum on the Hecke group $\mathbf H_4$. They are identical to the Markoff and Lagrange spectra of the unit circle. The Markoff spectrum on $\mathbf H_4$ is also known as the Markoff spectrum of index 2 sublattices by Vulakh and the Markoff spectrum of 2-minimal forms or $C$-minimal forms by Schmidt. They characterized the spectrum up to the first accumulation point, independently. We show that, after the first accumulation point, both spectra have positive Hausdorff dimension. Then we find gaps in the spectra and give a bound on Hall's ray. |
| title | The Markoff and Lagrange spectra on the Hecke group H4 |
| topic | Number Theory Dynamical Systems 11J06, 11J70 |
| url | https://arxiv.org/abs/2206.05441 |