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Multi-recurrence and van der Waerden systems

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Kraków, 30-348, Poland

    Dominik Kwietniak

  2. Department of Mathematics, Shantou University, Shantou, 515063, China

    Jian Li

  3. Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, 30-059, Poland

    Piotr Oprocha

  4. National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Ostrava, 70103, Czech Republic

    Piotr Oprocha

  5. Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    XiangDong Ye

Authors
  1. Dominik Kwietniak
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  2. Jian Li
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  3. Piotr Oprocha
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  4. XiangDong Ye
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Correspondence to Jian Li.

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Kwietniak, D., Li, J., Oprocha, P. et al. Multi-recurrence and van der Waerden systems. Sci. China Math. 60, 59–82 (2017). https://doi.org/10.1007/s11425-015-0860-8

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