Nilspace factors for general uniformity seminorms, cubic exchangeability and limits / Pablo Candela, Balázs Szegedy.

Format:

Book

Author/Creator:

Candela, Pablo, author.

Szegedy, Balazs, author.

Series:

Memoirs of the American Mathematical Society ; no. 1425.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1425

Language:

English

Subjects (All):

Nilpotent groups.

Curves, Cubic.

Ergodic theory.

Measure-preserving transformations.

Physical Description:

v, 101 pages : illustrations ; 26 cm.

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:

Chapter 1. Introduction

Chapter 2. Measure-theoretic preliminaries

Chapter 3. Cubic couplings

Chapter 4. The structure theorem for cubic couplings

Chapter 5. On characteristic factors associated with nilpotent group actions

Chapter 6. On cubic exchangeability

Chapter 7. Limits of functions on compact nilspaces.

Notes:

"July 2023, volume 287, number 1425 (third of 6 numbers)."

Includes bibliographical references (pages 99-101).

ISBN:

9781470465483

1470465485

OCLC:

1393204342