On irrationality measure functions

For a real $\alpha$ the ordinary irrationality measure function is defined as
$$ \psi_{\alpha}(t)\,=\,\min_{1\le q\le t, \, q\in \mathbb{Z}}||q\alpha ||,\quad t\,\ge\,1 $$
(here $||\xi || = \min_{a\in \mathbb{Z}}|\xi - a|$ is the distance from $\xi$ to the nearest integer). This function is connected with the best approximations to $\alpha$. Many Diophantine properties of real numbers can be described in terms of the irrationality measure function $\psi_\alpha (t)$. In particular it is convenient to define Lagrange and Dirichlet spectra in terms of the values
$$ \liminf_{t\to\infty} t\psi_\alpha (t)\,\,\,\, \text{and}\,\,\,\, \limsup_{t\to\infty} t\psi_\alpha (t). $$
Another interesting result is related to oscillation property of the difference $\psi_\alpha (t) -\psi_\beta (t).$
In our lecture, we will discuss certain properties of the functions
$$ \psi_\alpha^{[2]}(t)\,=\,\min_{
\begin{array}{c} (q,p): \, q,p\in \mathbb{Z}, 1\le q\le t, \cr (p,q) \neq (p_n, q_n) \,\forall\, n =0,1,2,3,... \end{array}
} |q\alpha -p| $$
and
$$ \psi_\alpha^{[2]*} (t)\,=\,\min_{
\begin{array}{c} (q,p): \, q,p\in \mathbb{Z}, 1\le q\le t, \cr p/q \neq p_n/q_n \,\forall\, n =0,1,2,3,... \end{array}
} |q\alpha -p| $$
related to the “second best” approximations and certain properties of the function $\mu_{\alpha}(t)$ associated with the Minkowski diagonal continued fraction for $\alpha$.