New examples of $M\setminus L$: intruder sets

TL;DR

This paper introduces new regions in the Markov and Lagrange spectra featuring intruder sets, discovered through computational search, and highlights their distinct combinatorial properties and the use of a flexible self-replication method.

Contribution

It presents novel examples of intruder sets in the spectra, expanding understanding of their structure with a modified self-replication approach.

Findings

Identification of new regions with intruder sets in $M\setminus L$

Use of computational methods to discover these regions

Introduction of a flexible self-replication technique

Abstract

We exhibit new examples of regions of $M\setminus L$ where $M$ and $L$ denote the Markov and Lagrange spectra, respectively. These regions have a different nature from all known regions studied so far: they contain \emph{intruder sets} associated with distinct combinatorics that trespass the region where self-replication holds. Our construction follows the usual self-replication method but replaces the standard local uniqueness condition with a more flexible and weaker property. These examples emerged from a large-scale computational search for regions of $M\setminus L$, which indicates that many such regions with intruder sets exist. We conclude with some open problems about these new regions.