New gaps on the Lagrange and Markov spectra

TL;DR

This paper identifies new gaps in the Lagrange and Markov spectra, improves bounds on the Hausdorff dimension of their difference, and demonstrates infinitely many maximal gaps accumulating to a known gap, using computational and renormalisation methods.

Contribution

It introduces new gaps in the spectra, refines the lower bound on the Hausdorff dimension of their difference, and proves the existence of infinitely many maximal gaps accumulating to a specific known gap.

Findings

New gaps near 3.938 in the spectra

Largest known elements of M\L identified

Lower bound of 0.593 on Hausdorff dimension of M\L

Abstract

Let $L$ and $M$ denote the Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. In this work, we exhibit new gaps of $L$ and $M$ using two methods. First, we derive such gaps by describing a new portion of $M\setminus L$ near to 3.938: this region (together with three other candidates) was found by investigating the pictures of $L$ recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of $M\setminus L$ and we improve upon a lower bound on the Hausdorff dimension of $M\setminus L$ obtained by the last two authors together with M. Pollicott and P. Vytnova (heuristically, we get a new lower bound of $0.593$ on the dimension of $M\setminus L$). Secondly, we use a renormalisation idea and a thickness criterion (reminiscent from the third author's PhD thesis) to detect infinitely many maximal gaps of $M$ accumulating to Freiman's gap preceding the so-called Hall's ray $[4.52782956616...,\infty)\subset L$.