Revisiting the nilpotent polynomial Hales-Jewett theorem

Abstract

Answering a question posed by Bergelson and Leibman in [6], we establish a nilpotent version of the polynomial Hales-Jewett theorem that contains the main theorem in [6] as a special case. Important to the formulation and the proof of our main theorem is the notion of a relative syndetic set (relative with respect to a closed non-empty subsets of $\beta \mathbf{G}$) [25]. As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions.

arXiv commentAdded journal biblio and addendum to ancillary files; 20 pages, 2nd revised version