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

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