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arXiv · 2412.05858

The Dirichlet spectrum

Alon Agin·Barak Weiss
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Abstract

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.

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Alon Agin, Barak Weiss. 2024-12-08. The Dirichlet spectrum. https://arxiv.org/abs/2412.05858

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