On the classical Lagrange and Markov spectra: new results on the local dimension and the geometry of the difference set

TL;DR

This paper explores the fractal geometry of the classical Lagrange and Markov spectra, analyzing their local Hausdorff dimensions, intersections, and the structure of their difference set, revealing new insights and maximal gaps.

Contribution

It introduces a detailed analysis of the local Hausdorff dimension of the spectra, characterizes their behavior on specific intervals, and identifies the largest known elements and gaps in the difference set.

Findings

Constructed intervals where the local dimension function is non-decreasing.

Proved the coincidence of intersections of derivatives of the spectra within these intervals.

Identified the largest known elements of the difference set and described new maximal gaps.

Abstract

Let $L$ and $M$ denote the classical Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. Inspired by three questions asked by the third author in previous work investigating the fractal geometric properties of the Lagrange and Markov spectra, we investigate the function $d_{loc}(t)$ that gives the local Hausdorff dimension at a point $t$ of $L'$. Specifically, we construct several intervals (having non-trivial intersection with $L'$) on which $d_{loc}$ is non-decreasing. We also prove that the respective intersections of $M'$ and $M''$ with these intervals coincide. Furthermore, we completely characterize the local dimension of both spectra when restricted to those intervals. Finally, we demonstrate the largest known elements of the difference set $M\setminus L$ and describe two new maximal gaps of $M$ nearby.