Multiple recurrence and convergence results associated to $${\text{F}}_P^\omega $$ -actions

Abstract

Using an ergodic inverse theorem obtained in our previous paper, we obtain limit formulae for multiple ergodic averages associated with the action of \({\text{F}}_p^\omega = \oplus {{\text{F}}_p}\). From this we deduce multiple Khintchine-type recurrence results analogous to those for ℤ-systems obtained by Bergelson, Host, and Kra, and also present some new counterexamples in this setting.