An Inverse Theorem for the Uniformity Seminorms Associated with the Action of $${{\mathbb {F}^{\infty}_{p}}}$$

Let $${\mathbb {F}}$$ a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action $${(T_{g})_{{g} \in \mathbb {F}^{\omega}}}$$ of the infinite abelian group $${\mathbb {F}^{\omega}}$$ on a probability space $${X = (X, \mathcal {B}, \mu)}$$ is generated by phase polynomials $${\phi : X \to S^{1}}$$ of degree less than C(k) on X, where C(k) depends only on k. In the case where $${k \leq {\rm char}(\mathbb {F})}$$ we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for $${\mathbb {Z}}$$ by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case $${k \leq {\rm char}(\mathbb {F})}$$ , with a partial result in low characteristic