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.
