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 t1 such that the portion of the Markov spectrum (−∞,t1)∩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. In addition, we also give a plot of the dimension function, which hasn't appeared previously in the literature to our knowledge.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.