A nilpotent IP polynomial multiple recurrence theorem

TL;DR

This paper extends multiple recurrence theorems in ergodic theory to nilpotent groups, combining polynomial mappings and combinatorial tools to generalize previous results in the field.

Contribution

It introduces a generalized nilpotent IP-polynomial multiple recurrence theorem, unifying and extending prior theorems by Bergelson, McCutcheon, and Leibman.

Findings

Generalization of Leibman's result on polynomial mappings into nilpotent groups

Development of a multiparameter nilpotent Hales-Jewett theorem

Establishment of a nilpotent IP-polynomial multiple recurrence theorem

Abstract

We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.