Dirichlet improvability in $L_p$-norms
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909637748457472 |
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| author | Moshchevitin, Nikolay Shulga, Nikita |
| author_facet | Moshchevitin, Nikolay Shulga, Nikita |
| contents | For a norm $F$ on $\mathbb{R}^2$, we consider the set of $F$-Dirichlet improvable numbers $\mathbf{DI}_F$. In the most important case of $F$ being an $L_p$-norm with $p=\infty$, which is a supremum norm, it is well-known that $\mathbf{DI}_F = \mathbf{BA}\cup \mathbb{Q}$, where $\mathbf{BA}$ is a set of badly approximable numbers. It is also known that $\mathbf{BA}$ and each $\mathbf{DI}_F$ are of measure zero and of full Hausdorff dimension.
Using classification of critical lattices for unit balls in $L_p$, we provide a complete and effective characterization of $\mathbf{DI}_p:=\mathbf{DI}_{F^{[p]}}$ in terms of the occurrence of patterns in regular continued fraction expansions, where $F^{[p]}$ is an $L_p$-norm with $p\in[1,\infty)$. This yields several corollaries. In particular, we resolve two open questions by Kleinbock and Rao by showing that the set $\mathbf{DI}_{p}\setminus \mathbf{BA}$ is of full Hausdorff dimension, as well as proving some results about the size of the difference $\mathbf{DI}_{p_1}\setminus \mathbf{DI}_{p_2}$. To be precise, we show that the set difference of Dirichlet improvable numbers in Euclidean norm ($p=2$) minus Dirichlet improvable numbers in taxicab norm ($p=1$) and vice versa, that is $\mathbf{DI}_{2}\setminus \mathbf{DI}_{1}$ and $\mathbf{DI}_{1}\setminus \mathbf{DI}_{2}$, are of full Hausdorff dimension.
We also find all values of $p$, for which the set $\mathbf{DI}_p^c\cap\mathbf{BA}$ has full Hausdorff dimension.
Finally, our characterization result implies that the number $e$ satisfies $e\in \mathbf{DI}_p$ if and only if $p\in(1,2)\cup(p_0,\infty)$ for some special constant $p_0\approx2.57$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06200 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirichlet improvability in $L_p$-norms Moshchevitin, Nikolay Shulga, Nikita Number Theory Dynamical Systems 11J83, 11H06, 11J70 For a norm $F$ on $\mathbb{R}^2$, we consider the set of $F$-Dirichlet improvable numbers $\mathbf{DI}_F$. In the most important case of $F$ being an $L_p$-norm with $p=\infty$, which is a supremum norm, it is well-known that $\mathbf{DI}_F = \mathbf{BA}\cup \mathbb{Q}$, where $\mathbf{BA}$ is a set of badly approximable numbers. It is also known that $\mathbf{BA}$ and each $\mathbf{DI}_F$ are of measure zero and of full Hausdorff dimension. Using classification of critical lattices for unit balls in $L_p$, we provide a complete and effective characterization of $\mathbf{DI}_p:=\mathbf{DI}_{F^{[p]}}$ in terms of the occurrence of patterns in regular continued fraction expansions, where $F^{[p]}$ is an $L_p$-norm with $p\in[1,\infty)$. This yields several corollaries. In particular, we resolve two open questions by Kleinbock and Rao by showing that the set $\mathbf{DI}_{p}\setminus \mathbf{BA}$ is of full Hausdorff dimension, as well as proving some results about the size of the difference $\mathbf{DI}_{p_1}\setminus \mathbf{DI}_{p_2}$. To be precise, we show that the set difference of Dirichlet improvable numbers in Euclidean norm ($p=2$) minus Dirichlet improvable numbers in taxicab norm ($p=1$) and vice versa, that is $\mathbf{DI}_{2}\setminus \mathbf{DI}_{1}$ and $\mathbf{DI}_{1}\setminus \mathbf{DI}_{2}$, are of full Hausdorff dimension. We also find all values of $p$, for which the set $\mathbf{DI}_p^c\cap\mathbf{BA}$ has full Hausdorff dimension. Finally, our characterization result implies that the number $e$ satisfies $e\in \mathbf{DI}_p$ if and only if $p\in(1,2)\cup(p_0,\infty)$ for some special constant $p_0\approx2.57$. |
| title | Dirichlet improvability in $L_p$-norms |
| topic | Number Theory Dynamical Systems 11J83, 11H06, 11J70 |
| url | https://arxiv.org/abs/2408.06200 |