# Admissible endpoints of gaps in the Lagrange spectrum

## TL;DR

This paper characterizes which numbers in the Lagrange spectrum are admissible, providing a necessary and sufficient condition, and constructs an infinite series of numbers that are not admissible.

## Contribution

It establishes a complete criterion for admissibility in the Lagrange spectrum and identifies infinitely many non-admissible numbers.

## Key findings

- Derived a necessary and sufficient condition for admissibility.
- Constructed an infinite series of non-admissible numbers.
- Enhanced understanding of the structure of the Lagrange spectrum.

## Abstract

We call a positive real number $\lambda$ admissible if it belongs to the Lagrange spectrum and there exists an irrational number $\alpha$ such that $\mu(\alpha)=\lambda$. Here $\mu(\alpha)$ denotes the Lagrange constant of $\alpha$ - maximal real number $c$ such that $\forall \varepsilon>0$ the inequality $|\alpha-\frac{p}{q}|\le\frac{1}{(c-\varepsilon)q^2}$ has infinitely many solutions for relatively prime $p$ and $q$. In this paper we establish a necessary and sufficient condition of admissibility of the Lagrange spectrum element and construct an infinite series of not admissible numbers.