An inverse theorem for the uniformity seminorms associated with the action of $F^ω$

Details An inverse theorem for the uniformity seminorms associated with the action of $F^ω$ An inverse theorem for the uniformity seminorms associated with the action of $F^ω$

2009-01-17

arxiv.org/abs/0901.2602v2

Geom. Funct. Anal. 19 (2010), No. 6, 1539-1596

Mathematics

Dynamical Systems

59 pages, 2 figures, to appear, GAFA. Referee suggestions incorporated. Also, the paper has been shortened at the request of t

Scientific paper

10.1007/s00039-010-0051-1

Let $\F$ a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm $U^k(\X)$ for an ergodic action $(T_g)_{g \in \F^\omega}$ of the infinite abelian group $\F^\omega$ on a probability space $X = (X,\B,\mu)$ 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 $k \leq \charac(\F)$ we obtain the sharp result $C(k)=k$. This is a finite field counterpart of an analogous result for $\Z$ by Host and Kra. In a companion paper 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 $k \leq \charac(\F)$, with a partial result in low characteristic.

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