On the Folklore set and Dirichlet spectrum for matrices
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916613276565504 |
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| author | Hussain, Mumtaz Schleischitz, Johannes Ward, Benjamin |
| author_facet | Hussain, Mumtaz Schleischitz, Johannes Ward, Benjamin |
| contents | We study the Folklore set of Dirichlet improvable matrices in $\mathbb R^{m\times n}$ which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs $m,n$ apart from $\{m,n\}=\{ 1,1\}$ and $\{m,n\}=\{ 2,3\}$ in a constructive manner. For a wide range of integer pairs $(m,n)$ we construct subsets of the Folklore set with an exact prescribed Dirichlet constant (in some right neighbourhood of $0$). This enables us to provide information on the Dirichlet Spectrum of matrices. The key technique of our construction is to build first vectors of a given Diophantine type, and then to show that most `liftings' to matrices will preserve this Diophantine type. This is a variant of a method introduced by Moshchevitin for uniform approximation. Our technique is often also applicable to arbitrary norms. As a corollary, we obtain lower bounds on the Hausdorff dimension of these sets. These statements complement previous results of the middle-named author (Selecta Math. 2023), Beresnevich et. al. (Adv. Math. 2023), and Das et. al. (Adv. Math. 2024). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13451 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Folklore set and Dirichlet spectrum for matrices Hussain, Mumtaz Schleischitz, Johannes Ward, Benjamin Number Theory Classical Analysis and ODEs Dynamical Systems We study the Folklore set of Dirichlet improvable matrices in $\mathbb R^{m\times n}$ which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs $m,n$ apart from $\{m,n\}=\{ 1,1\}$ and $\{m,n\}=\{ 2,3\}$ in a constructive manner. For a wide range of integer pairs $(m,n)$ we construct subsets of the Folklore set with an exact prescribed Dirichlet constant (in some right neighbourhood of $0$). This enables us to provide information on the Dirichlet Spectrum of matrices. The key technique of our construction is to build first vectors of a given Diophantine type, and then to show that most `liftings' to matrices will preserve this Diophantine type. This is a variant of a method introduced by Moshchevitin for uniform approximation. Our technique is often also applicable to arbitrary norms. As a corollary, we obtain lower bounds on the Hausdorff dimension of these sets. These statements complement previous results of the middle-named author (Selecta Math. 2023), Beresnevich et. al. (Adv. Math. 2023), and Das et. al. (Adv. Math. 2024). |
| title | On the Folklore set and Dirichlet spectrum for matrices |
| topic | Number Theory Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2402.13451 |