Hausdorff dimension of Gauss--Cantor sets and two applications to classical Lagrange and Markov spectra

Carlos Matheus, Carlos Gustavo Moreira, Mark Pollicott, Polina Vytnova

Published 2021-06-11Version 1

This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem \ref{t.A}, provides a rigorous estimate on the smallest value $t_1$ such that the portion of the Markov spectrum $(-\infty,t_1)\cap M$ has Hausdorff dimension $1$. Our second result, Theorem \ref{t.B}, gives a new upper bound on the Hausdorff dimension of the set difference $M\setminus L$. Our method combines new facts about the structure of the classical spectra together with finer estimates on the Hausdorff dimension of Gauss--Cantor sets of continued fraction expansions whose entries satisfy appropriate restrictions.