  The main goal of this paper is to obtain optimal estimates on the speed of
equidistribution of nilflows on higher step nilmanifolds. Under a Diophantine
condition on the frequencies of the toral projection of the flow, we prove that
for almost all points on the nilmanifold orbits become equidistributed at
polynomial speed with exponent which decays quadratically as a function of the
number of steps. The main novelty is the introduction of new techniques of
renormalization (rescaling) in absence of a truly recurrent renormalization
dynamics. Quantitative equidistribution estimates are derived from bounds on
the scaling of invariant distributions (in Sobolev norms) and on the geometry
of the nilmanifold under the rescaling.
