Host–Kra theory for -systems and multiple recurrence

2021/01/31 by Or Shalom
Mathematics · #Advanced Topology and Set Theory #Combinatorics #Countable set #Discrete mathematics #Ergodic theory #Geometry #Inverse #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Mathematics #Multiset #Omega #Physics #Prime (order theory) #Pure mathematics #Quantum mechanics #math.DS

paper · pdf · doi:10.1017/etds.2021.109

65 pages. Added an acknowledgement. This is the final accepted version appeared in Ergodic Theory and Dynamical Systems. arXiv admin note: substantial text overlap with arXiv:2006.06470

openalex publication_date 2021/10/25 · openalex created_date 2021/11/08 · arxiv created 2022/02/14 · arxiv updated 2022/02/15 · openalex updated_date 2026/08/05

Abstract

Abstract Let \mathcal P be an (unbounded) countable multiset of primes (that is, every prime may appear multiple times) and let G=\bigoplus _p∈ \mathcal P\mathbb F_p . We develop a Host–Kra structure theory for the universal characteristic factors of an ergodic G -system. More specifically, we generalize the main results of Bergelson, Tao and Ziegler [An inverse theorem for the uniformity seminorms associated with the action of \mathbb F_p^∞ . Geom. Funct. Anal. 19 (6) (2010), 1539–1596], who studied these factors in the special case \mathcal P=\p,p,p,… \ for some fixed prime p . As an application we deduce a Khintchine-type recurrence theorem in the flavor of Bergelson, Tao and Ziegler [Multiple recurrence and convergence results associated to F_p^ω -actions. J. Anal. Math. 127 (2015), 329–378] and Bergelson, Host and Kra [Multiple recurrence and nilsequences. Invent. Math. 160 (2) (2005), 261–303, with an appendix by I. Ruzsa].

Citations

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