We will discuss some recent results on the fractal geometry of the Markov and Lagrange spectra, M and L which are classical objects from the theory of Diophantine approximations, and their set difference. In particular, we discuss recent results in collaboration with Erazo, Gutiérrez-Romo and Romaña, which give precise asymptotic estimates for the fractal dimensions of the Markov and Lagrange spectra near 3 (their smaller accumulation point) and other recent collaborations with Jeffreys, Matheus, Pollicott and Vytnova, in which we prove that the Hausdorff dimension of the complement of the Lagrange spectrum in the Markov spectrum has Hausdorff dimension between 0.594561 and 0.796445. Finally we will discuss a recent work in collaboration with H. Erazo, D. Lima, C. Matheus and S. Vieira in which we prove that inf(M\L)=3.

We will relate these results to symbolic dynamics, continued fractions and to the study of the fractal geometry of arithmetic sums of regular Cantor sets, a subject also important for the study of homoclinic bifurcations in Dynamical Systems.

Carlos Gustavo Moreira，现任巴西纯粹与应用数学研究所（IMPA）教授。Moreira教授是巴西科学院（ABC）院士，第三世界科学院（TWAS）院士，巴西数学奥林匹克委员会（BMOC）成员。Moreira教授主要研究动力系统，遍历理论，数论以及组合数学。在丢番图逼近，分型几何学等具体研究领域做出了重要贡献。他曾于1989年获得国际数学奥林匹克（IMO）铜牌，1990年获得了国际数学奥林匹克（IMO）金牌，2009年获得了UMALCA 奖，2010年获得了TWAS奖，2018年获得国家数学竞赛世界联盟(WFNMC)颁发的保罗.厄多斯奖（Paul Erdös Award）。他也是 2018年国际数学家大会（ICM）60分钟报告人。