On effective equidistribution for higher step nilflows

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.