The inverse conjecture for the Gowers norms for finite-dimensional vector spaces over a finite field asserts, roughly speaking, that a bounded function has large Gowers norm if and only if it correlates with a phase polynomial of degree at most , thus is a polynomial of degree at most . In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler. In low characteristic we obtain a partial result, in which the phase polynomial is allowed to be of some larger degree . The full inverse conjecture remains open in low characteristic; the counterexamples found so far in this setting can be avoided by a slight reformulation of the conjecture.
"The inverse conjecture for the Gowers norm over finite fields via the correspondence principle." Anal. PDE 3 (1) 1 - 20, 2010. https://doi.org/10.2140/apde.2010.3.1