Hausdorff dimension of the Lagrange and Markov spectra near 3

Rodolfo Gutierrez-Romo (University of Chile)

Monday 27th February 16:00-17:00 Maths 311B


The Lagrange spectrum L and Markov spectrum M are fractal subsets of the real line related to Diophantine properties of irrational numbers or of indefinite quadratic forms. Their geometry is very complicated and has been studied since the seminal papers of Markov in 1879–1880.

Moreira showed in the 2018 that the Hausdorff dimensions dim(L ∩ (-∞, t)) and dim(M ∩ (-∞, t)) are equal. Letting d(t) = dim(L ∩ (-∞, t)) = dim(M ∩ (-∞, t)), it is known that d(t) = 0 for t ≤ 3, and Moreira also showed that that d(t) > 0 for every t > 3. In this talk, we will discuss how d(t) behaves near t = 3 and, in particular, show that d(3 + exp(-r)) behaves roughly like log(r)/r for large enough r > 0.

This is joint work with Harold Erazo, Carlos Gustavo Moreira and Sergio Romaña.

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