Cube spaces and the multiple term return times theorem

TL;DR

This paper presents a new proof of Rudolph's multiple term return times theorem using Host-Kra structure theory, extending the result to arbitrary F{46}lner sequences and nilsequences.

Contribution

It introduces a novel proof method based on Host-Kra structure theory, providing characteristic factors and generalizing to broader contexts.

Findings

Provides a new proof of Rudolph's theorem

Extends the theorem to arbitrary F{46}lner sequences

Establishes a Wiener-Wintner-type result for nilsequences

Abstract

We give a new proof of Rudolph's multiple term return times theorem based on Host-Kra structure theory. Our approach provides characteristic factors for all terms, works for arbitrary tempered F{\o}lner sequences and also yields a multiple term Wiener-Wintner-type return times theorem for nilsequences.