LEADER 05493nam 22006733 450 001 9910915792003321 005 20240723124741.0 010 $a1-4704-7541-3 035 $a(MiAaPQ)EBC30671912 035 $a(Au-PeEL)EBL30671912 035 $a(PPN)272106526 035 $a(CKB)27902413600041 035 $a(EXLCZ)9927902413600041 100 $a20230804d2023 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aNilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2023. 210 4$d©2023. 215 $a1 online resource (114 pages) 225 1 $aMemoirs of the American Mathematical Society Series ;$vv.287 311 08$aPrint version: Candela, Pablo Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits Providence : American Mathematical Society,c2023 9781470465483 327 $aCover -- 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 ?-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 ?. -- 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. 330 $a"We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host- Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host-Kra type seminorms for measure-preserving actions of countable nilpotent groups. This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability. These are sequences indexed by the infinite discrete cube, such that for every integer k [geq] 0 the joint distribution's marginals on affine subcubes of dimension k are all equal. In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society Series 606 $aNilpotent groups 606 $aCurves, Cubic 606 $aErgodic theory 606 $aMeasure-preserving transformations 606 $aDynamical systems and ergodic theory -- Ergodic theory$2msc 606 $aDynamical systems and ergodic theory -- Ergodic theory -- General groups of measure-preserving transformations$2msc 606 $aProbability theory and stochastic processes$2msc 606 $aProbability theory and stochastic processes -- Probability theory on algebraic and topological structures$2msc 606 $aNumber theory -- Sequences and sets -- Arithmetic combinatorics; higher degree uniformity$2msc 615 0$aNilpotent groups. 615 0$aCurves, Cubic. 615 0$aErgodic theory. 615 0$aMeasure-preserving transformations. 615 7$aDynamical systems and ergodic theory -- Ergodic theory. 615 7$aDynamical systems and ergodic theory -- Ergodic theory -- General groups of measure-preserving transformations. 615 7$aProbability theory and stochastic processes. 615 7$aProbability theory and stochastic processes -- Probability theory on algebraic and topological structures. 615 7$aNumber theory -- Sequences and sets -- Arithmetic combinatorics; higher degree uniformity. 676 $a512/.2505 676 $a512.2505 686 $a37Axx$a37A15$a60-XX$a60Bxx$a11B30$2msc 700 $aCandela$b Pablo$01779783 701 $aSzegedy$b Balázs$01779784 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910915792003321 996 $aNilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits$94303340 997 $aUNINA