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

Dynamical Systems