Metastability in the Furstenberg-Zimmer tower

TL;DR

This paper reveals that the Furstenberg-Katznelson proof of Szemerédi's theorem relies on a weaker property of the maximal distal factor, which appears early in the transfinite construction at the ^{^{}} level, rather than its full strength.

Contribution

It demonstrates that the proof only depends on a combinatorial weakening of the maximal distal factor, appearing at a low transfinite level, simplifying the structural requirements.

Findings

The proof relies on a weaker property of the maximal distal factor.

This property appears at the ^{^{}} level in the transfinite hierarchy.

The full strength of the maximal distal factor is not necessary for the proof.

Abstract

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 Szemer\'edi'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.