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

Part of: Measure-theoretic ergodic theory Ergodic theory

Published online by Cambridge University Press:  25 October 2021

OR SHALOM
OR SHALOM*
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Jerusalem 91904, Israel
*
e-mail: Or.Shalom@mail.huji.ac.il
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Abstract

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].

Keywords

characteristic factorsnilsystemscocycles

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators
Secondary: 28D05: Measure-preserving transformations 37A45: Relations with number theory and harmonic analysis
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 1 , January 2023 , pp. 299 - 360
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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