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arXiv · 2106.06572

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

Carlos Matheus·Carlos Gustavo Moreira·Mark Pollicott·Polina Vytnova
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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.

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Carlos Matheus, Carlos Gustavo Moreira, Mark Pollicott, Polina Vytnova. 2022-08-30. Hausdorff dimension of Gauss--Cantor sets and two applications to classical Lagrange and Markov spectra. https://arxiv.org/abs/2106.06572

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