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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.

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber Δ euc . In addition to the metric length it contains information on the direction of the segment. In this paper we study restrictions on the Δ euc -valued side lengths of polygons in Euclidean buildings. The main result is that for thick Euclidean buildings X the set $${\mathcal{P}n(X)}$$ of possible Δ euc -valued side lengths of oriented n-gons depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of Δ euc -valued weights of semistable weighted configurations on the Tits boundary ∂ Tits X.The side lengths of polygons in symmetric spaces of noncompact type are studied in the related paper [KLM1]. Applications of the geometric results in both papers to algebraic group theory are given in [KLM2].

The purpose of the paper is to present new estimates on incomplete character sums in finite fields that are of the strength of Burgess’ celebrated theorem for prime fields. More precisely, an inequality of this type is obtained in $${F_{p^2}}$$ and also for binary quadratic forms, improving on the work of Davenport–Lewis and on several results due to Burgess. The arguments are based on new estimates for the multiplicative energy of certain sets that allow us to improve the amplification step in Burgess’ method.