Abstract:
Akhunzhanov and Shatskov defined the Dirichlet spectrum, corresponding to m×n matrices and to norms on Rm and Rn. In case (m,n) = (2,1) and using the Euclidean norm on R2, they showed that the spectrum is an interval. We generalize this result to arbitrary (m,n) with max(m,n)>1 and arbitrary norms, improving previous works from recent years. We also define some related spectra and show that they too are intervals. We also prove the existence of matrices exhibiting special properties with respect to their uniform exponent. Our argument is a modification of an argument of Khintchine from 1926.
