Dirichlet spectrum for one linear form

TL;DR

This paper characterizes the Dirichlet spectrum for a single linear form in n-dimensional space, establishing it as the entire interval [0,1], thus advancing understanding in Diophantine approximation.

Contribution

The paper determines the Dirichlet spectrum for one linear form with respect to the maximum norm, extending previous results and providing a foundation for future research on systems of forms.

Findings

Dirichlet spectrum for one linear form is [0,1]

Improves on recent work by Beresnevich et al.

Sets groundwork for general systems of linear forms.

Abstract

For $n\geq 2$, we determine the Dirichlet spectrum in $\Rn$ with respect to a linear form and the maximum norm as the entire interval $[0,1]$. This natural result improves on recent work of Beresnevich, Guan, Marnat, Ram\'irez and Velani, and complements a subsequent paper by the author where the analogous result was proved for simultaneous approximation. Various generalisations that can be obtained by similar methods as in the latter paper are indicated. We believe that our results are an important step towards resolving the very open analogous problem for a general system of linear forms.