Revisiting the Nilpotent Polynomial Hales-Jewett Theorem

Abstract

Answering a question posed by Bergelson and Leibman in [7], we establish a nilpotent version of the Polynomial Hales-Jewett Theorem that is a common generalization of both the main theorem in [7] and the Polynomial Hales-Jewett Theorem (PHJ). Important to the formulation and the proof of our main theorem is Shuungula's, Zelenyuk's, and Zelenyuk's [30] notion of a relative syndetic set (relative with respect to two closed non-empty subsets of \(\beta G\)). Using their language we first reformulate the classical PHJ as a theorem about abelian groups before we proceed to generalize it to nilpotent groups. As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions. We also offer a reformulation of the Density Hales-Jewett Theorem based on an extension of the notion of relative syndeticity to a relative version of upper and lower Banach density.