Fractal geometry of the complement of Lagrange spectrum in Markov spectrum

  • Carlos Matheus

    École Polytechnique, Palaiseau, France
  • Carlos Gustavo Moreira

    Nankai University, Tianjin, China and IMPA, Rio de Janeiro, Brazil
Fractal geometry of the complement of Lagrange spectrum in Markov spectrum cover
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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 MLM\setminus L of the Lagrange spectrum LL in the Markov spectrum MM remained somewhat mysterious. In fact, it was shown by G. Freiman (in 1968 and 1973) and M. Flahive (in 1977) that MLM\setminus L contains infinite countable subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements of MLM\setminus L were <12=3.46< \sqrt{12} = 3.46\ldots, and this was the status quo of our knowledge of MLM\setminus L until 2017.

In this article, we show the following two results. First, we prove that MLM\setminus L 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 MLM\setminus L 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. 593–633

DOI 10.4171/CMH/498