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

TL;DR

This paper investigates the Hausdorff dimension of Gauss--Cantor sets and applies these findings to analyze the classical Lagrange and Markov spectra, providing new bounds and estimates for their structure.

Contribution

It introduces new estimates on the Hausdorff dimension related to the Markov spectrum and the difference between Markov and Lagrange spectra, using refined analysis of continued fractions.

Findings

Estimated the smallest value t_1 for the Markov spectrum with Hausdorff dimension 1

Provided a new upper bound on the Hausdorff dimension of M \\ L

Combined structural properties of spectra with Gauss--Cantor set analysis

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.