Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits / Pablo Candela and Balázs Szegedy.

Format:

Book

Author/Creator:

Candela, Pablo, author.

Szegedy, Balazs, author.

Series:

Memoirs of the American Mathematical Society ; Volume 287.

Memoirs of the American Mathematical Society Series ; Volume 287

Language:

English

Subjects (All):

Curves, Cubic.

Measure-preserving transformations.

Nilpotent groups.

Physical Description:

1 online resource (114 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host- Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host-Kra type seminorms for measure-preserving actions of countable nilpotent groups. This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability. These are sequences indexed by the infinite discrete cube, such that for every integer k [geq] 0 the joint distribution's marginals on affine subcubes of dimension k are all equal. In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Acknowledgments

Chapter 2. Measure-theoretic preliminaries

2.1. Some basic notions

2.2. Couplings

2.3. Closed properties in a coupling space

2.4. Localization

2.5. Conditional independence in set lattices

2.6. Idempotent couplings

Chapter 3. Cubic couplings

3.1. Conditional independence of simplicial sets

3.2. Tricubes

3.3. ^{ }-convolutions and ^{ }-seminorms associated with a cubic coupling

3.4. Fourier -algebras

3.5. Properties of ^{ }-convolutions

3.6. Topologization of cubic couplings

3.7. Continuous ⁿ-convolutions

3.8. Topological nilspace factors of \ns

Chapter 4. The structure theorem for cubic couplings

4.1. Verifying the ergodicity and composition axioms

4.2. Complete dependence of corner couplings

4.3. Convolution neighbourhoods

4.4. Construction of the coupling Υ.

4.5. Verifying the corner-completion axiom

Chapter 5. On characteristic factors associated with nilpotent group actions

Chapter 6. On cubic exchangeability

Chapter 7. Limits of functions on compact nilspaces

Appendix A. Background results from measure theory

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Candela, Pablo Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits

ISBN:

9781470475413

1470475413