Discrete part of the second Lagrange spectrum

Given an irrational number $\alpha$ consider its irrationality measure function ${\psi }_{\alpha }(t)=\underset{1\le q\le t,q\in \mathbb{Z}}{\min }\mathrm{\parallel }q\alpha \mathrm{\parallel }$. The set of all values of $\lambda (\alpha )=(\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}t{\psi }_{\alpha }(t){)}^{-1}$ where $\alpha$ runs through the set $\mathbb{R}\setminus \mathbb{Q}$ is called the Lagrange spectrum $\mathbb{L}$. In a paper by Moshchevitin an irrationality measure function ${\psi }_{\alpha }^{[2]}(t)=\underset{1\le q\le t,q\in \mathbb{Z},q\mathrm{\ne }{q}_i}{\min }\mathrm{\parallel }q\alpha \mathrm{\parallel }$ was introduced. In other words, we consider the best approximations by fractions, whose denominators are not the denominators of the convergents to $\alpha$. Replacing the function ${\psi }_{\alpha }$ in the definition of $\mathbb{L}$ by ${\psi }_{\alpha }^{[2]}$, one can get a set ${\mathbb{L}}_2$ which is called the ‘‘second’’ Lagrange spectrum. In this paper we give the complete structure of discrete part of ${\mathbb{L}}_2$.