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On higher-order Fourier analysis in characteristic p

Part of: Representation theory of groups Sequences and sets

Published online by Cambridge University Press:  27 January 2023

PABLO CANDELA ,
DIEGO GONZÁLEZ-SÁNCHEZ  and
BALÁZS SZEGEDY
PABLO CANDELA*
Affiliation:
Universidad Autónoma de Madrid and ICMAT, Madrid 28049, Spain
DIEGO GONZÁLEZ-SÁNCHEZ
Affiliation:
MTA Alfréd Rényi Institute of Mathematics, Budapest H-1053, Hungary (e-mail: diegogs@renyi.hu, szegedyb@gmail.com)
BALÁZS SZEGEDY
Affiliation:
MTA Alfréd Rényi Institute of Mathematics, Budapest H-1053, Hungary (e-mail: diegogs@renyi.hu, szegedyb@gmail.com)
*
e-mail: pablo.candela@uam.es
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Abstract

In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb {F}_p$, with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups. The applications include a description of the Host–Kra factors of ergodic $\mathbb {F}_p^\omega $-systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for $k\leq p+1$ the kth Host–Kra factor is an Abramov system of order at most k, extending a result of Bergelson–Tao–Ziegler that holds for $k< p$. We illustrate the utility of p-homogeneous nilspaces also by showing that the structure theorem yields a new proof of the Tao–Ziegler inverse theorem for Gowers norms on $\mathbb {F}_p^n$.

Keywords

Gowers normsfinite fieldsHost--Kra factorsnilspaces

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 20D15: Nilpotent groups, $p$-groups
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 12 , December 2023 , pp. 3971 - 4040
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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