Informal Group Action Seminar

The classical Lagrange and Markov spectra encode the features of certain Diophantine approximations problems. The first systematic study of these spectra was done by Markov in 1880. Since then, these objects attracted the attention of several authors (including Hurwitz, Frobenius, Perron, ...) for many reasons: for instance, these spectra are dynamically meaningful because they describe the cusp excursions of geodesics of the modular surface.

In 1968, Freiman exhibited near 3.11 an infinite countable subset of the complement M\L of the Lagrange spectrum L in the Markov spectrum M. After that, the works of Freiman in 1973 and Flahive in 1977 showed that M\L also contains an infinite countable subset near 3.29. On the other hand, Cusick conjectured in 1975 that L and M should coincide after \sqrt{12}.

In this talk, we will discuss some joint works with Moreira exploiting certain dynamical ideas showing that M\L is richer than it was initially thought by Cusick, Flahive and Freiman. In particular, we will show that Cusick's conjecture is not true and, moreover, the Hausdorff dimension of M\L is located between 0.531 and 0.888.