On two-dimensional Dirichlet spectrum

TL;DR

This paper introduces the two-dimensional Dirichlet spectrum based on the Euclidean norm, characterizes it explicitly as the interval [0, 2/√3], and explores its properties in Diophantine approximation.

Contribution

It defines the two-dimensional Dirichlet spectrum and proves that it equals the interval [0, 2/√3], providing a complete characterization.

Findings

The spectrum D_2 is exactly [0, 2/√3].

The spectrum is a closed interval.

The result advances understanding of multidimensional Diophantine approximation.

Abstract

We define two-dimensional Dirichlet spectrum (with respect to Euclidean norm) as D_2=\lambda\in\mathbf{R} | \exists \mathbf{v}=(v_1,v_2)\in \mathbf {R}^2: \limsup\limits_{t\rightarrow\infty} {t\cdot\psi_{v}^2(t)}=\lambda, where \psi_{v}(t)=\min\limits_{1\leqslant q\leqslant t}\sqrt{|q v_1|^2+|q v_2|^2} is the two-dimensional "irrationality measure function". Our main result states the equality D_2=[0; 2/sqrt{3}].