Analysis of two step nilsequences

TL;DR

This paper investigates the properties and classification of two-step nilsequences, extending classical almost periodic sequence theory to this broader class within ergodic theory and additive combinatorics.

Contribution

It establishes fundamental properties and a classification scheme for two-step nilsequences, broadening the understanding of their structure and applications.

Findings

Basic properties of 2-step nilsequences are proven.

A classification scheme for 2-step nilsequences is developed.

Connections to classical almost periodic sequences are clarified.

Abstract

Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg's proof of Szemer\'edi's Theorem and have since played a role in problems in additive combinatorics. Nilsequences are a generalization of almost periodic sequences and we study which portions of the classical theory for almost periodic sequences can be generalized for two step nilsequences. We state and prove basic properties for 2-step nilsequences and give a classification scheme for them.