The Dirichlet spectrum

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 $(m,n)\ne (1,1)$ 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.