  We prove an effective equidistribution result for a class of higher step
nilflows, called filiform nilflows, and derive bounds on Weyl sums for higher
degree polynomials with a power saving comparable to the best known, derived by
J. Bourgain, C. Demeter and L. Guth and by T. Wooley from their proof of
Vinogradov Main Conjecture. Our argument is based on ideas from dynamical
systems (cohomological equations, invariant distributions) and on non-Abelian
harmonic analysis.
