Metastability in the Furstenberg–Zimmer tower

Volume 210 / 2010

Jeremy Avigad, Henry Towsner Fundamenta Mathematicae 210 (2010), 243-268 MSC: 37A05, 03F03, 03E15. DOI: 10.4064/fm210-3-2


According to the Furstenberg–Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemerédi's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg–Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the $\omega^{\omega^\omega}$th level.


  • Jeremy AvigadDepartment of Philosophy
    and Department of Mathematical Sciences
    Carnegie Mellon University
    Pittsburgh, PA 15213, U.S.A.
  • Henry TowsnerDepartment of Mathematics
    University of California
    Los Angeles, CA 90095-1555, U.S.A.

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