\(\mathrm{IP}_r^\ast\)-recurrence and nilsystems
type: Item
instance of: scholarly article
title: \(\mathrm{IP}_r^\ast\)-recurrence and nilsystems
zbMATH Open document ID: 1444.37012
published in: Advances in Mathematics
publication date: 2018-10-29T00:00:00Z
review text: The paper deals with the notion of \(\mathrm{IP}^{*}\)-recurrence introduced by \textit{H. Furstenberg} [Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey: Princeton University Press (1981; Zbl 0459.28023)]. The authors show that translations on compact nilmanifolds (which are known to be distal) are characterized by a stronger property of \(\mathrm{IP}^{*}_r\)-recurrence. It is also proved that the property of \(\mathrm{IP}^{*}_r\)-recurrence is equivalent to ostensibly, a much stronger property of polynomial \(\mathrm{IP}^{*}_r\)-recurrence. This should be juxtaposed with the fact that for general distal transformations the polynomial \(\mathrm{IP}^{*}\)-recurrence is strictly stronger than the \(\mathrm{IP}^{*}\)-recurrence.
Mathematics Subject Classification ID: 37B20
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 37B02
zbMATH DE Number: 6967039
zbMATH Keywords: nilsystems
zbMATH Keywords: sets of recurrence
zbMATH Keywords: generalized polynomial mappings
zbMATH Keywords: finite IP-sets
author: V. Bergelson
author: A. Leibman
reviewed by: Juan L. G. Guirao
MaRDI profile type: Publication
full work available at URL: j.aim.2018.09.032
OpenAlex ID: W2963641471
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DOI: 10.1016/J.AIM.2018.09.032
description: scientific article; zbMATH DE number 6967039