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 th level.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm210-3-2,
author = {Jeremy Avigad and Henry Towsner},
title = {Metastability in the Furstenberg-Zimmer tower},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {243-268},
zbl = {1230.37004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm210-3-2}
}
Jeremy Avigad; Henry Towsner. Metastability in the Furstenberg-Zimmer tower. Fundamenta Mathematicae, Tome 209 (2010) pp. 243-268. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm210-3-2/