The Dirichlet spectrum with respect to $L_1$ norm is $\left[\frac12,1\right]$

Abstract

We prove that the one-dimensional Dirichlet spectrum with respect to approximation in $L_1$ norm $\mathbb{D}^{[1]}$ satisfies $$ \mathbb{D}^{[1]}=\left[\frac12,1\right]. $$ This is equivalent to the fact that the Minkowski spectrum $\mathbb M$, associated with the Minkowski diagonal continued fraction, satisfies $$ \mathbb M=\left[\frac14,\frac12\right]. $$ Further, we show that level sets $$ Θ_m=\{α\in(0,1)\setminus\mathbb Q:\mathfrak m(α)=m\}, $$ where $\mathfrak{m}(α)$ is the Minkowski constant of $α$, have Hausdorff dimension strictly greater than $1/2$ for any $m\in(1/4,1/2]$, while $\dim_H Θ_{1/4}=\frac{1}{2}$.