An inverse theorem for the uniformity seminorms associated with the action of $F$ _p^∞.

Vitaly Bergelson, Terence Tao and Tamar Ziegler. An inverse theorem for the uniformity seminorms associated with the action of $F$ _p^∞. GAFA Vol19-6 (2009), 1539-1596.

Abstract

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 of the infinite abelian group $F$ ^∞ on a probability space X = (X,B,μ) is generated by phase polynomials φ: X → S¹ 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. 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 ≤ Char(F), with a partial result in low characteristic.

Blog links:
An inverse theorem for the uniformity seminorms associated with the action of F^infty_p « What’s new [What’s new@terrytao.wordpress.com/2009]
The Inverse Conjecture for the Gowers Norms « In Theory [In Theory@lucatrevisan.wordpress.com/2008]
Cohomology for dynamical systems « What’s new [What’s new@terrytao.wordpress.com/2008]
Some notes on "non-classical" polynomials in finite characteristic « What’s new [What’s new@terrytao.wordpress.com/2008]

Online Videos:
An inverse theorem associated with the action of

F

_p^∞. [MSRI Introduction to Ergodic Theory and Additive Combinatorics workshop].
An inverse theorem for the Gowers norms over finite fields [IAS Workshop on Pseudorandomness in Mathematical Structures].

An inverse theorem for the uniformity seminorms associated with the action of Fp∞.

Abstract

An inverse theorem for the uniformity seminorms associated with the action of $F$ _p^∞.