$HD(M\setminus L)<0.986927$

TL;DR

This paper demonstrates that certain parts of the complement of the Lagrange spectrum within the Markov spectrum can be represented as sums of Cantor sets with controlled dimensions, establishing that this complement has Hausdorff dimension less than one.

Contribution

It provides a new geometric and dimensional analysis of the complement of the Lagrange spectrum in the Markov spectrum using Cantor sets.

Findings

$Mackslash L$ has Hausdorff dimension less than one

Portions of $Mackslash L$ are subsets of sums of Cantor sets

The Hausdorff dimension of these sets is explicitly controlled

Abstract

We show that several portions of the complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ can be seen as subsets of arithmetic sums of Cantor sets with controlled Hausdorff dimensions. In particular, we prove that $M\setminus L$ has Hausdorff dimension strictly smaller than one.