#### Vol. 8, No. 1, 2019

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Admissible endpoints of gaps in the Lagrange spectrum

### Dmitry Gayfulin

Vol. 8 (2019), No. 1, 47–56
DOI: 10.2140/moscow.2019.8.47
##### Abstract

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

${\mu }^{-1}\left(\alpha \right)=\underset{p\in ℤ,\phantom{\rule{0.3em}{0ex}}q\in ℕ}{liminf}|q\left(q\alpha -p\right)|.$

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

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

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.

##### Keywords
Lagrange spectrum, Diophantine approximation, continued fractions
Primary: 11J06