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 (−∞, t 1 ) ∩ 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 \ 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.