Giovanni Forni (Université de Maryland) : "Effective ergodicity of higher step nilflows and bounds on Weyl sums"

Giovanni Forni

Université de Maryland

Exposé  réel 

 Lieu : Salle Visio, M3

 Résumé : The best known upper bounds on Weyl sumsfor higher degree polynomials were first derived by Bourgain, Demeter and Guth, from their proof (2015) of ''Vinogradov  Main Conjecture'', based on ''decoupling'',  and later matched by T. Wooley (2019) by a refinement of his method of ''efficient congruencing''.
In joint work with L. Flaminio we gave a direct proof (2014) of a similar bound for Weyl sumsbased 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 with L. Flaminio.