Fractal geometry of the complement of Lagrange spectrum in Markov spectrum
Carlos MatheusÉcole Polytechnique, Palaiseau, France
Carlos Gustavo MoreiraNankai University, Tianjin, China and IMPA, Rio de Janeiro, Brazil
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 of the Lagrange spectrum in the Markov spectrum remained somewhat mysterious. In fact, it was shown by G. Freiman (in 1968 and 1973) and M. Flahive (in 1977) that contains infinite countable subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements of were , and this was the status quo of our knowledge of until 2017.
In this article, we show the following two results. First, we prove that is richer than it was previously thought because it contains a Cantor set of Hausdorff dimension larger than 1/2 near 3.7: in particular, this solves (negatively) Cusick's conjecture mentioned above. Secondly, we show that is not very thick: its Hausdorff dimension is strictly smaller than one.
Cite this article
Carlos Matheus, Carlos Gustavo Moreira, Fractal geometry of the complement of Lagrange spectrum in Markov spectrum. Comment. Math. Helv. 95 (2020), no. 3, pp. 593–633DOI 10.4171/CMH/498