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Host–Kra theory for $\bigoplus _{p\in P}{\mathbb {F}}_p$ -systems and multiple recurrence

Ergodic theory and dynamical systems, 2023-01, Vol.43 (1), p.299-360 [Peer Reviewed Journal]

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Title:
Host–Kra theory for $\bigoplus _{p\in P}{\mathbb {F}}_p$ -systems and multiple recurrence
Author: SHALOM, OR
Subjects: Original Article
Is Part Of: Ergodic theory and dynamical systems, 2023-01, Vol.43 (1), p.299-360
Description: Let $\mathcal {P}$ be an (unbounded) countable multiset of primes (that is, every prime may appear multiple times) and let $G=\bigoplus _{p\in \mathcal {P}}\mathbb {F}_p$ . 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^\infty $ . Geom. Funct. Anal. 19(6) (2010), 1539–1596], who studied these factors in the special case $\mathcal {P}=\{p,p,p,\ldots \}$ 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^\omega $ -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].
Publisher: Cambridge, UK: Cambridge University Press
Language: English
Identifier: ISSN: 0143-3857
EISSN: 1469-4417
DOI: 10.1017/etds.2021.109
Source: AUTh Library subscriptions: ProQuest Central

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