Equidistribution of nilflows and bounds on Weyl sums

TL;DR

This paper establishes effective equidistribution results for certain nilflows and derives bounds on Weyl sums for higher degree polynomials, combining dynamical systems and harmonic analysis techniques.

Contribution

It introduces new bounds on Weyl sums for higher degree polynomials using methods from nilflow dynamics and non-Abelian harmonic analysis.

Findings

Effective equidistribution for filiform nilflows

Bounds on Weyl sums with power saving

Methods applicable to higher degree polynomial sums

Abstract

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.