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

 Download this article For screen For printing  Recent Issues Volume 10, Issue 1 Volume 9, Issue 4 Volume 9, Issue 3 Volume 9, Issue 2 Volume 9, Issue 1 Volume 8, Issue 4 Volume 8, Issue 3 Volume 8, Issue 2 Volume 8, Issue 1  The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement founded and published with the scientific support and advice of the Moscow Institute of Physics and Technology ISSN (electronic): 2640-7361 ISSN (print): 2220-5438 Previously Published Author Index To Appear Other MSP Journals  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
##### Milestones
Received: 10 January 2018
Accepted: 17 March 2018
Published: 11 August 2018
##### Authors
 Dmitry Gayfulin Steklov Mathematical Institute Russian Academy of Sciences Moscow Russia