Quantitative ergodic theory and Diophantine approximation

The fundamental problem of Diophantine approximation is to quantify
the density of rational numbers in reals. We investigate this problem
on surfaces equipped with group actions, and establish analogues
of Khinchin and Jarnik theorems in this setting. The main ingredient
of the proof is an ergodic theorem with explicit rate determined by
the spectral gap in automorphic representations.
This is a joint work with A.Ghosh and A.Nevo.