Fractal geometry of the complement of Lagrange spectrum in Markov spectrum

2018/03/03 by Matheus, Carlos, Moreira, Carlos Gustavo
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1803.01230

Abstract

The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement M∖ L of the Lagrange spectrum L in the Markov spectrum M remained somewhat mysterious. In fact, it was shown by G. Freiman (in 1968 and 1973) and M. Flahive (in 1977) that M∖ L contains infinite countable subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements of M∖ L were

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