A consequence of the factorisation theorem for polynomial orbits on nilmanifolds

TL;DR

This paper refines a key theorem related to polynomial orbits on nilmanifolds, strengthening its conclusions to support advanced arithmetic applications and correcting previous oversights.

Contribution

It provides a generalized and strengthened version of a fundamental theorem on polynomial orbits, improving its applicability in arithmetic contexts.

Findings

Enhanced theorem applicable to polynomial orbits on nilmanifolds

Correction of previous proof oversight

Supports further arithmetic applications

Abstract

We discuss a consequence of Green and Tao's factorisation theorem for polynomial orbits on nilmanifolds, adjusted to the requirements of certain arithmetic applications. More precisely, we prove a generalisation of Theorem 16.4, Acta Arith. 154 (2012), 235-306, by slightly rearranging its proof. The thus achieved strengthening of the result removes an oversight in the above-cited paper which resulted from the previously too weak conclusion. Since this type of result proved essential for further applications, we take the opportunity to discuss it in more detail.