Séance Séminaire

Séminaire Gaston Darboux

Optimal Diophantine exponents on homogeneous varieties, using spectral methods and Sarnak's density hypothesis

We will begin by giving a brief introduction to problems of Diophantine approximation on homogeneous spaces of algebraic groups. We will then briefly describe some optimal results that can be deduced using harmonic analysis and dynamics, given a certain spectral assumption. We will describe Sarnak's density conjecture, and explain how a recently established version of it for quaternion groups can be utilized to establish - unconditionally - some optimal exponents for Diophantine approximation on the hyperbolic plane (and other homogeneous spaces). Partly based on joint work with Anish Ghosh and Alex Gorodnik, and partly based on joint work with Mikolaj Fraczyk and Alex Gorodnik.