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Multiple correlation sequences not approximable by nilsequences

Part of: Sequences and sets

Published online by Cambridge University Press:  16 July 2021

JOP BRIËT  and
BEN GREEN Open the ORCID record for BEN GREEN [Opens in a new window]
JOP BRIËT
Affiliation:
Centrum Wiskunde & Informatica (CWI), Science Park 123, 1098XG, Amsterdam, The Netherlands (e-mail: j.briet@cwi.nl)
BEN GREEN*
Affiliation:
Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Rd, OxfordOX2 6QW, UK
*
e-mail: ben.green@maths.ox.ac.uk
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Abstract

We show that there is a measure-preserving system $(X,\mathscr {B}, \mu , T)$ together with functions $F_0, F_1, F_2 \in L^{\infty }(\mu )$ such that the correlation sequence $C_{F_0, F_1, F_2}(n) = \int _X F_0 \cdot T^n F_1 \cdot T^{2n} F_2 \, d\mu $ is not an approximate integral combination of $2$ -step nilsequences.

Keywords

multiplerecurrencenilsequence

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 11B25: Arithmetic progressions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2711 - 2722
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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