inf(M \ L)=3

TL;DR

This paper investigates the divergence of the Lagrange and Markov spectra beyond 3, showing they differ immediately after this point and analyzing the Hausdorff dimension of their differences using advanced dynamical and number theoretic methods.

Contribution

It proves that the spectra differ immediately after 3 and provides lower bounds on the Hausdorff dimension of their difference sets using dynamical systems and number theory techniques.

Findings

L and M spectra coincide up to 3 but diverge immediately after.

The difference set has positive Hausdorff dimension near 3.

Lower bounds on the dimension ratio are established using probabilistic and number theoretic methods.

Abstract

The Lagrange and Markov spectra $L$ and $M$ describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed, $L\cap (0,3) = M\cap (0,3)$ is a discrete set of explicit quadratic irrationals accumulating only at $3$. In this article, we show that the statement above ceases to be true immediately after $3$: in particular, $L\cap (3,3+\varepsilon)\neq M\cap (3,3+\varepsilon)$ for all $\varepsilon>0$, and thus $\inf(M\setminus L)=3$. In fact, we derive this result as a by-product of lower bounds on the Hausdorff dimension of $(M\setminus L)\cap (3,3+\varepsilon)$ implying that $\liminf\limits_{\varepsilon\to 0} \frac{\dim_H((M\setminus L)\cap(3,3+\varepsilon))}{\dim_H(M\cap (3,3+\varepsilon))}\geq \frac{1}{2}$ and, as it turns out, these bounds are obtained from the study of projections of Cartesian products of almost affine dynamical Cantor sets via an argument of probabilistic flavor based on Baker--W\"ustholz theorem on linear forms in logarithms of algebraic numbers.