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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.20300 |
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| _version_ | 1866912352933249024 |
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| author | Villamil, Christian Camilo Silva Moreira, Carlos Gustavo |
| author_facet | Villamil, Christian Camilo Silva Moreira, Carlos Gustavo |
| contents | We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence $\{a_r\}_{r\in \mathbb{N}}$ converging to $3$, such that for any $r$ one can find a subset $\mathcal{B}_r\subset (a_{r+1},a_r)\cap \mathcal{L}^{'}$ with the property that the Hausdorff dimension of $((a_{r+1},a_r)\cap \mathcal{L})\setminus \mathcal{B}_r$ is less than the Hausdorff dimension of $\mathcal{B}_r$ and for $t\in \mathcal{B}_r$ the sets of irrational numbers with Lagrange value bounded by $t$ and exactly $t$ respectively, have the same Hausdorff dimension. We also show that, as $t$ varies in $\mathcal{B}_r$, this Hausdorff dimension is a strictly increasing function. Finally, in relation to $\mathcal{M}\setminus \mathcal{L}$, we find $C>0$ such that we can bound from above the Hausdorff dimension of $(\mathcal{M}\setminus \mathcal{L})\cap (-\infty,3+ρ)$ by $\frac{\log (\abs{\log ρ})-\log (\log(\abs{\log ρ}))+C}{\abs{\log ρ}}$ if $ρ>0$ is small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20300 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of some subsets of the Lagrange and Markov spectra near $3$ Villamil, Christian Camilo Silva Moreira, Carlos Gustavo Dynamical Systems We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence $\{a_r\}_{r\in \mathbb{N}}$ converging to $3$, such that for any $r$ one can find a subset $\mathcal{B}_r\subset (a_{r+1},a_r)\cap \mathcal{L}^{'}$ with the property that the Hausdorff dimension of $((a_{r+1},a_r)\cap \mathcal{L})\setminus \mathcal{B}_r$ is less than the Hausdorff dimension of $\mathcal{B}_r$ and for $t\in \mathcal{B}_r$ the sets of irrational numbers with Lagrange value bounded by $t$ and exactly $t$ respectively, have the same Hausdorff dimension. We also show that, as $t$ varies in $\mathcal{B}_r$, this Hausdorff dimension is a strictly increasing function. Finally, in relation to $\mathcal{M}\setminus \mathcal{L}$, we find $C>0$ such that we can bound from above the Hausdorff dimension of $(\mathcal{M}\setminus \mathcal{L})\cap (-\infty,3+ρ)$ by $\frac{\log (\abs{\log ρ})-\log (\log(\abs{\log ρ}))+C}{\abs{\log ρ}}$ if $ρ>0$ is small. |
| title | Hausdorff dimension of some subsets of the Lagrange and Markov spectra near $3$ |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2504.20300 |