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

The Furstenberg-Zimmer structure theorem for $\mathbb{Z}^d$ 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 Szemer\'edi's theorem, and Bergelson and Liebman further generalized to a polynomial Szemer\'edi'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, $\omega^{\omega^{\omega^\omega}}$).