Fractal geometry of the complement of Lagrange spectrum in Markov spectrum

Abstract

The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ remained somewhat mysterious. In fact, it was shown by G. Freiman (in 1968 and 1973) and M. Flahive (in 1977) that $M\setminus L$ contains infinite \emph{countable} subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements of $M\setminus L$ were $<\sqrt{12}=3.46\dots$, and this was the \emph{status quo} of our knowledge of $M\setminus L$ until 2017. In this article, we show the following two results. First, we prove that $M\setminus L$ is \emph{richer} than it was previously thought because it contains a Cantor set of Hausdorff dimension larger than $1/2$ near $3.7$: in particular, this solves (negatively) Cusick's conjecture mentioned above. Secondly, we show that $M\setminus L$ is \emph{not} very thick: its Hausdorff dimension is strictly smaller than one.