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Juxtaposing combinatorial and ergodic properties of large sets of integers

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Abstract

The classical approach of Furstenberg allows one to associate with any large set E ⊆ ℤ a dynamical system \({\mathbb{X}_E} = \left( {X,{\cal B},\mu ,T} \right)\) which “encodes” the combinatorial properties of E via the multiple recurrence properties of the transformation T. While one can always assume without loss of generality that \({\mathbb{X}_E}\) is ergodic, the requirement of ergodicity of T2 puts rather stringent combinatorial constraints on the set E. We undertake a close study of the connection between the combinatorial richness of large sets in ℤ and ergodic properties of the corresponding system \({\mathbb{X}_E}\). In particular, we characterize, in combinatorial terms, totally ergodic (resp. weakly mixing) sets E, i.e., sets for which T is totally ergodic (resp. weakly mixing). This leads to numerous new combinatorial applications.

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Authors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Vitaly Bergelson & Andreu Ferré Moragues

Authors
  1. Vitaly Bergelson
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  2. Andreu Ferré Moragues
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Corresponding author

Correspondence to Vitaly Bergelson.

Additional information

Dedicated to Benjy Weiss upon reaching g’vurot.

H. Furstenberg and his collaborators showed how ideas from dynamical systems could be used to obtain old and new results [of combinatorial nature]. The basic paradigm involved interpreting the object under investigation as a single orbit of an entire dynamical system. This study of patterns in large sets has developed into a broad discipline and has injected powerful new methods into basic combinatorial questions.

[In] single orbit dynamics … the main object of interest is a single orbit and its properties and the global dynamics is a tool. The reverse situation, in which the main object of interest is the global system and the individual orbit is but a tool is also not uncommon. We call this the principle of one for all… This means describing a global system as the orbit closure of a single orbit. This is a powerful tool in the construction of a variety of examples

—Benjamin Weiss, Single Orbit Dynamics, [We].

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Bergelson, V., Moragues, A.F. Juxtaposing combinatorial and ergodic properties of large sets of integers. Isr. J. Math. 251, 173–238 (2022). https://doi.org/10.1007/s11856-022-2441-0

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  • DOI: https://doi.org/10.1007/s11856-022-2441-0

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