Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911817377251328 |
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| author | Moreira, Carlos Gustavo Villamil, Christian Camilo Silva |
| author_facet | Moreira, Carlos Gustavo Villamil, Christian Camilo Silva |
| contents | We prove that for any $η$ that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets $k^{-1}((-\infty,η])$ and $k^{-1}(η)$, which are the sets of irrational numbers with best constant of Diophantine approximation bounded by $η$ and exactly $η$ respectively, have the same Hausdorff dimension. We also show that, as $η$ varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14646 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum Moreira, Carlos Gustavo Villamil, Christian Camilo Silva Dynamical Systems We prove that for any $η$ that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets $k^{-1}((-\infty,η])$ and $k^{-1}(η)$, which are the sets of irrational numbers with best constant of Diophantine approximation bounded by $η$ and exactly $η$ respectively, have the same Hausdorff dimension. We also show that, as $η$ varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function. |
| title | Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2309.14646 |