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

Multiple recurrence and convergence results associated to $\mathbb{F}_{p}^{\omega}$-actions