On the fractal geometry of the Lagrange and Markov spectra.

Wednesday, November 22, 2017
Ora Inizio: 
Ora Fine: 
Carlos Matheus
Paris XIII

After the remarkable works of Markov in 1879 and 1880, the Lagrange and Markov spectra (coding arithmetic properties of irrational numbers and indefinite binary quadratic forms) were studied by several authors (including Perron, Hall, Freiman, Cusick, Flahive, ...).

In this talk, we will discuss the complement of the Lagrange spectrum L in the Markov spectrum M. More precisely, after recalling the results of Freiman, Cusick and Flahive from the 70's and 80's showing that M\L contains a countable, infinite subset, we will show that the Hausdorff dimension of M\L is strictly between 0 and 1.

This is a joint work with Carlos Gustavo (Gugu) Moreira.