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

Abstract

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 2.1, 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 3.1, 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.

arXiv comment50 pages, 6 figures, 7 tables, an appendix with a pseudocode concerning the calculation of Hausdorff dimension of Gauss--Cantor sets. Final version (based on the referee report) to appear in Adv. Math