The Dirichlet spectrum
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913601077379072 |
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| author | Agin, Alon Weiss, Barak |
| author_facet | Agin, Alon Weiss, Barak |
| contents | Akhunzhanov and Shatskov defined the Dirichlet spectrum, corresponding to $m \times n$ matrices and to norms on $\mathbb{R}^m$ and $\mathbb{R}^n$. In case $(m,n) = (2,1)$ and using the Euclidean norm on $\mathbb{R}^2$, they showed that the spectrum is an interval. We generalize this result to arbitrary $(m,n) \neq (1,1)$ and arbitrary norms, improving previous works from recent years. We also define some related spectra and show that they too are intervals. Our argument is a modification of an argument of Khintchine from 1926. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05858 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dirichlet spectrum Agin, Alon Weiss, Barak Number Theory 11J06, 11J13, 11H06 Akhunzhanov and Shatskov defined the Dirichlet spectrum, corresponding to $m \times n$ matrices and to norms on $\mathbb{R}^m$ and $\mathbb{R}^n$. In case $(m,n) = (2,1)$ and using the Euclidean norm on $\mathbb{R}^2$, they showed that the spectrum is an interval. We generalize this result to arbitrary $(m,n) \neq (1,1)$ and arbitrary norms, improving previous works from recent years. We also define some related spectra and show that they too are intervals. Our argument is a modification of an argument of Khintchine from 1926. |
| title | The Dirichlet spectrum |
| topic | Number Theory 11J06, 11J13, 11H06 |
| url | https://arxiv.org/abs/2412.05858 |