It is know since the work of Furstenberg that the equidistribution of the fractional parts of polynomials sequences with irrational leading coeeficient can be derived from the unique ergodicity of (certain) nilflows. We will present some results on the speed of convergence of ergodic averages of nilflows under Diophantine conditions and discuss the relation with known results and conjectures on bounds of Weyl sums (exponential sums for polynomial sequences). The method of proof is based on the analysis of the action of a suitable 'renormalization' on the space invariant distributions for nilflows (in particular it makes no use of number theory). The content of this talk is joint work with L. Flaminio.