Markov spectrum near Freiman's isolated points in $M\setminus L$

TL;DR

This paper analyzes the structure of the Markov spectrum near Freiman's isolated points, computes the smallest known element outside the Lagrange spectrum, and estimates the Hausdorff dimension of a key subset.

Contribution

It describes the structure of the Markov spectrum in a specific interval, computes the smallest element outside the Lagrange spectrum, and provides a lower bound for its Hausdorff dimension.

Findings

Identified the structure of the Markov spectrum in the interval (c_infinity, C_infinity).

Computed the smallest known element of M extbackslash L.

Estimated the Hausdorff dimension of a subset of the spectrum as > 0.2628.

Abstract

Freiman proved in 1968 that the Lagrange and Markov spectra do not coincide by exhibiting a countable infinite collection $\mathcal{F}$ of isolated points of the Markov spectrum which do not belong the Lagrange spectrum. In this paper, we describe the structure of the elements of the Markov spectrum in the largest interval $(c_{\infty}, C_{\infty})$ containing $\mathcal{F}$ and avoiding the Lagrange spectrum. In particular, we compute the smallest known element $f$ of $M\setminus L$, and we show that the Hausdorff dimension of the portion of the Markov spectrum between $c_{\infty}$ and $C_{\infty}$ is $> 0.2628$.