Polynomial sequences in discrete nilpotent groups of step 2

TL;DR

This paper investigates polynomial sequences in step 2 nilpotent groups, establishing boundedness of maximal functions and singular integrals, and proving pointwise convergence of ergodic averages using novel analytical tools like a nilpotent circle method.

Contribution

It introduces a nilpotent circle method and analytical techniques for studying polynomial sequences in nilpotent groups, advancing understanding in non-commutative harmonic analysis.

Findings

Boundedness of maximal functions and singular integrals in nilpotent groups

Almost everywhere convergence of ergodic averages along polynomial sequences

Development of a nilpotent circle method for analytical purposes

Abstract

We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular, we present what we call a "nilpotent circle method" that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.