AN INVERSE THEOREM FOR THE UNIFORMITY SEMINORMS ASSOCIATED WITH THE ACTION OF F-p(infinity)

摘要

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 is an element of F omega) of the infinite abelian group F-omega on a probability space X = (X, beta, mu) is generated by phase polynomials phi : X -> S-1 of degree less than C(k) on X, where C(k) depends only on k. In the case where k <= char(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 [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 <= char(F), with a partial result in low characteristic.