Admissible endpoints of gaps in 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 $\forall \epsilon >0$ the inequality $|\alpha -\frac{p}{q}|\le \frac{1}{(c-\epsilon )q^2}$ 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.

arXiv comment10 pages

References in corpus (1)

• Attainable numbers and the Lagrange spectrum