The Dirichlet spectrum

TL;DR

This paper generalizes the Dirichlet spectrum for matrices with arbitrary dimensions and norms, proving that these spectra are intervals, extending previous results and using a modified classical argument.

Contribution

It extends the known interval property of the Dirichlet spectrum to all dimensions except (1,1) and arbitrary norms, broadening the scope of earlier findings.

Findings

Dirichlet spectrum is an interval for arbitrary (m,n) ≠ (1,1)

Generalization to arbitrary norms on b^m and b^n

Related spectra are also shown to be intervals

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.