Multi-recurrence and van der Waerden systems
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We explore recurrence properties arising from dynamical approach to the van der Waerden theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.
Keywordsmulti-recurrent points van der Waerden systems multiple recurrence theorem multiple IP-recurrence property multi-non-wandering points
MSC(2010)37B20 37B05 37A25 05D10
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This work was supported by the National Science Centre (Grant No. DEC-2012/07/E/ST1/00185), National Natural Science Foundation of China (Grant Nos. 11401362, 11471125, 11326135, 11371339 and 11431012), Shantou University Scientific Research Foundation for Talents (Grant No. NTF12021) and the Project of LQ1602 IT4Innovations Excellence in Science. The authors thank Jie Li for the careful reading and helpful comments. A substantial part of this paper was written when the authors attended the activity “Dynamics and Numbers”, June–July 2014, held at the Max Planck Institute f¨ur Mathematik (MPIM) in Bonn, Germany. Some part of the work was continued when the authors attended a conference held at the Wuhan Institute of Physics and Mathematics, China, and the satellite conference of 2014 ICM at Chungnam National University, South Korea. The authors are grateful to the organizers for their hospitality.
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