### Effective ergodicity of nilflows and bound on Weyl sums

Ergodic Theory and Dynamical Systems Seminar

6th October 2022, 2:00 pm – 3:00 pm

Fry Building, G07

The best known upper bounds on Weyl sums for 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 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.

This is joint work with L. Flaminio.

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