Metastability and the Furstenberg-Zimmer Tower II: Polynomial and Multidimensional Szemeredi's Theorem

2009/09/30 by Henry Towsner, Towsner, Henry
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #Mathematical Dynamics and Fractals #math.DS #math.LO

paper · pdf · doi:10.48550/arxiv.0909.5668

arxiv created 2009/09/30 · openalex publication_date 2009/09/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Furstenberg-Zimmer structure theorem for ℤ^d actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemerédi's theorem, and Bergelson and Liebman further generalized to a polynomial Szemerédi's theorem. Beleznay and Foreman showed that, in general, this tower can have any countable height. Here we show that these proofs do not require the full height of this tower; we define a weaker combinatorial property which is sufficient for these proofs, and show that it always holds at fairly low levels in the transfinite construction (specifically, ω^ω^ωω).

Related

• Metastability in the Furstenberg-Zimmer tower
• Deducing the multidimensional Szemeredi Theorem from an infinitary removal lemma
• A multi-dimensional Szemerédi theorem for the primes via a correspondence principle
• Bounds in a popular multidimensional nonlinear Roth theorem
• The ergodic and combinatorial approaches to Szemerédi's theorem