Uniformity in the Wiener-Wintner theorem for nilsequences

TL;DR

This paper extends the Wiener-Wintner theorem to nilsequences, providing a uniform version and a topological extension using Fourier analysis and Sobolev embeddings, advancing understanding in ergodic theory.

Contribution

It introduces a uniform extension of the Wiener-Wintner theorem for nilsequences and a topological version, utilizing Fourier analysis and Sobolev embeddings.

Findings

Established a uniform Wiener-Wintner theorem for nilsequences.

Extended the topological Wiener-Wintner theorem to nilsequence settings.

Applied Fourier analysis and Sobolev embeddings in the proofs.

Abstract

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.