The Dirichlet spectrum with respect to $L_1$ norm is $\left[\frac{1}{2},1\right]$

Abstract

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

29 pages including one Appendix. Comments are welcome