The best known upper bound on Weyl sums for higher degree polynomials were derived by Bourgain,
Demeter and Guth from their proof (2015) of Vinogradov ''Main Conjecture'' (on the number of integral solutions
of certain Diophantine equations). A similar, slightly weaker, bound follows from T. Wooley's series of results
(in the 2010's) on the ''Main Conjecture''.
In joint work with L. Flaminio we gave a direct proof (2014) of a similar bound for Weyl sums based on ideas from
dynamical systems (cohomological equations for nilflows, renormalization), but only for a full measure set of lower
degree coefficients (vs. for all of them). We will outline our approach and a recent improvement which allows us
to eliminate this shortcoming and to give a new proof of the Bourgain-Demeter-Guth bound. In fact, our proof works
under a new multidimensional Diophantine condition which appears to be weaker than classical one-dimensional
conditions in the above mentioned works. This is joint work, in progress, with L. Flaminio. | 
