Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3
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| Format: | Preprint |
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2025
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| author | Moreira, Carlos Gustavo Erazo, Harold Angelini, Nicolas |
| author_facet | Moreira, Carlos Gustavo Erazo, Harold Angelini, Nicolas |
| contents | We study the generalized Hausdorff dimension of some natural subsets of $k^{-1}(3)$, where $k^{-1}(3)$ consists of the real numbers $x$ for which $\left| x-\frac{p}{q} \right|<\frac{1}{(3+\varepsilon)q^2}$ has infinitely many rational solutions $\frac{p}{q}$ for any $\varepsilon<0$ but only finitely many for any $\varepsilon>0$. It is well known that $k^{-1}(3)$ is an uncountable set with Hausdorff dimension zero. Given any dimension function $h$, we determine the exact "cut point" at which the generalized Hausdorff dimension $\mathcal{H}^h(k^{-1}(3))$ drops from infinity to zero. In particular we show that such a measure is always zero or not $σ$--finite, and, as an application, we can classify topologically $k^{-1}(3)$. Moreover, we show that the subset of attainable elements of $k^{-1}(3)$ has the same generalized Hausdorff dimension as $k^{-1}(3)$, but the subset of non--attainable elements of $k^{-1}(3)$ has a "strictly smaller" generalized Hausdorff dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10328 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3 Moreira, Carlos Gustavo Erazo, Harold Angelini, Nicolas Number Theory Dynamical Systems 11J06, 11J70, 28A78 We study the generalized Hausdorff dimension of some natural subsets of $k^{-1}(3)$, where $k^{-1}(3)$ consists of the real numbers $x$ for which $\left| x-\frac{p}{q} \right|<\frac{1}{(3+\varepsilon)q^2}$ has infinitely many rational solutions $\frac{p}{q}$ for any $\varepsilon<0$ but only finitely many for any $\varepsilon>0$. It is well known that $k^{-1}(3)$ is an uncountable set with Hausdorff dimension zero. Given any dimension function $h$, we determine the exact "cut point" at which the generalized Hausdorff dimension $\mathcal{H}^h(k^{-1}(3))$ drops from infinity to zero. In particular we show that such a measure is always zero or not $σ$--finite, and, as an application, we can classify topologically $k^{-1}(3)$. Moreover, we show that the subset of attainable elements of $k^{-1}(3)$ has the same generalized Hausdorff dimension as $k^{-1}(3)$, but the subset of non--attainable elements of $k^{-1}(3)$ has a "strictly smaller" generalized Hausdorff dimension. |
| title | Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3 |
| topic | Number Theory Dynamical Systems 11J06, 11J70, 28A78 |
| url | https://arxiv.org/abs/2511.10328 |