On some Diophantine spectra

Let $\alpha$ be an irrational number, and let
$$ \psi_\alpha(t)=\min_{\mathbb{Z}_+\ni q\le t}\|q\alpha\| $$
be the function of measure of its irrationality. In the talk, we discuss some old and new results concerning Lagrange spectrum
$$ \mathbb{L}=\Bigl\{\lambda\in\mathbb{R}:\exists\,\alpha\in\mathbb{R}\setminus\mathbb{Q}\ \liminf_{t\to\infty}t\psi_\alpha(t)=\lambda\Bigr\}, $$
Dirchlet spectrum
$$ \mathbb{D} = \{ d\in \mathbb{R}:\,\, \exists \alpha \in \mathbb{R}\setminus\mathbb{Q}\,\,\, \limsup_{t\to \infty} t\psi_\alpha (t) = d\}, $$
and the spectrum
$$ \mathbb{M}=\Bigl\{m\in\mathbb{R}:\exists\,\alpha\in\mathbb{R}\setminus\mathbb{Q}\ \limsup_{t\to\infty}t\mu_\alpha(t)=m\Bigr\}, $$
connected with the function $\mu_\alpha(t)$, arising in the analysis of Minkowski diagonal fraction.