$M\setminus L$ near 3

TL;DR

This paper constructs new elements in the Markov spectrum outside the Lagrange spectrum, forming a decreasing sequence converging to 3, suggesting that the difference set might be non-closed and answering a longstanding open question.

Contribution

The authors explicitly construct four new elements of M\L in distinct components and provide evidence that an infinite sequence of such elements converges to 3, challenging previous assumptions about the spectra.

Findings

Constructed four new elements in M\L near 3

Presented evidence for an infinite decreasing sequence in M\L converging to 3

Indicated that M\L may not be closed, answering Bousch's question

Abstract

We construct four new elements $3.11>m_1>m_2>m_3>m_4$ of $M\backslash L$ lying in distinct connected components of $\mathbb{R}\setminus L$, where $M$ is the Markov spectrum and $L$ is the Lagrange spectrum. These elements are part of a decreasing sequence $(m_k)_{k\in\mathbb{N}}$ of elements in $M$ converging to $3$ and we give some evidence towards the possibility that $m_k\in M\setminus L$ for all $k\geq 1$. In particular, this indicates that $3$ might belong to the closure of $M\setminus L$, so that the answer to Bousch's question about the closedness of $M\setminus L$ might be negative.