$M\backslash L$ is not closed

TL;DR

The paper demonstrates that the set difference between the Markov and Lagrange spectra is not closed by showing a specific point in the Lagrange spectrum is accumulated by points outside it, answering a question negatively.

Contribution

It proves that the complement of the Lagrange spectrum within the Markov spectrum is not closed, providing a counterexample to a question posed by T. Bousch.

Findings

$1+3/ oot2$ is in Lagrange spectrum

Points outside Lagrange spectrum accumulate at a point in L

The set difference M extbackslash L is not closed

Abstract

We show that $1+3/\sqrt{2}$ is a point of the Lagrange spectrum $L$ which is accumulated by a sequence of elements of the complement $M\setminus L$ of the Lagrange spectrum in the Markov spectrum $M$. In particular, $M\setminus L$ is not a closed subset of $\mathbb{R}$, so that a question by T. Bousch has a negative answer.