Distribution and quantitative density of orbits

Abstract: We will be interested in the distribution of orbits for group actions
on homogeneous spaces. This, in particular, involves analysis of
the discrepancy functions and the density exponent. We explain
an analytic method that allows to estimate these quantities.
It turns out that establishing optimal estimates is closely related
to a deep conjecture about the spectral gap property, which is currently out of reach.
Nonetheless, we sketch a more refined approach involving
estimates on the density exceptions to this conjecture that
allows to establish the optimal density unconditionally.