Abstract
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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bergelson, V., Tao, T. & Ziegler, T. An Inverse Theorem for the Uniformity Seminorms Associated with the Action of \({{\mathbb {F}^{\infty}_{p}}}\) . Geom. Funct. Anal. 19, 1539–1596 (2010). https://doi.org/10.1007/s00039-010-0051-1
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DOI: https://doi.org/10.1007/s00039-010-0051-1