Admissible endpoints of gaps in the Lagrange spectrum

Abstract

For any irrational number $\alpha$ define the Lagrange constant $\mu \left(\alpha \right)$ by

$${\mu }^{-1}\left(\alpha \right)=\underset{p\in \mathbb{Z},q\in \mathbb{N}}{liminf}\left|q\left(q\alpha -p\right)\right|.$$

The set of all values taken by $\mu \left(\alpha \right)$ as $\alpha$ varies is called the Lagrange spectrum $\mathbb{L}$. An irrational $\alpha$ is called attainable if the inequality

$$\left|\alpha -\frac{p}{q}\right|⩽\frac{1}{\mu \left(\alpha \right)q^2}$$

holds for infinitely many integers $p$ and $q$. We call a real number $\lambda \in \mathbb{L}$ admissible if there exists an irrational attainable $\alpha$ such that $\mu \left(\alpha \right)=\lambda$. In a previous paper we constructed an example of a nonadmissible element in the Lagrange spectrum. In the present paper we give a necessary and sufficient condition for admissibility of a Lagrange spectrum element. We also give an example of an infinite sequence of left endpoints of gaps in $\mathbb{L}$ which are not admissible.