Given an irrational number $\alpha$ consider its irrationality measure function $\psi_{ \alpha}(t)=\min\limits_{ 1\le q\le t, q\in\mathbb{ Z}}\|q\alpha\|$. The set of all values of $\lambda(\alpha)=(\limsup\limits_{ t\to\infty} t\psi_{ \alpha}(t))^{ -1}$ 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 $\psi^{ [2]}_{ \alpha}(t)=\min\limits_{ 1\le q\le t, q\in\mathbb{ Z},q\ne q_i}\|q\alpha\|$ 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 $\psi^{ [2]}_{ \alpha}$, 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 $\mathbb{ L}_2$.
Categories: math.NT