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- Title
Fractal geometry of the complement of Lagrange spectrum in Markov spectrum.
- Authors
Matheus, Carlos; Moreira, Carlos Gustavo
- 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 objectswere first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement M n 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 n L contains infinite countable subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements ofM n L were < √12 = 3:46 ..., and this was the status quo of our knowledge of M n L until 2017. In this article, we show the following two results. First, we prove thatM nL 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 M n L is not very thick: its Hausdorff dimension is strictly smaller than one.
- Subjects
MARKOV spectrum; LAGRANGE spectrum; FRACTALS; FRACTAL dimensions; NUMBER theory; CANTOR sets; DIOPHANTINE approximation
- Publication
Commentarii Mathematici Helvetici, 2020, Vol 95, Issue 3, p593
- ISSN
0010-2571
- Publication type
Article
- DOI
10.4171/CMH/498