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arXiv · 1704.08060

Admissible endpoints of gaps in the Lagrange spectrum

Dmitry Gayfulin
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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.

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Dmitry Gayfulin. 2017-04-26. Admissible endpoints of gaps in the Lagrange spectrum. https://doi.org/10.2140/moscow.2019.8.47

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