  The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional
vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a
bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it
correlates with a phase polynomial $\phi = e_\F(P)$ of degree at most $d-1$,
thus $P: V \to \F$ is a polynomial of degree at most $d-1$. In this paper, we
develop a variant of the Furstenberg correspondence principle which allows us
to establish this conjecture in the large characteristic case $\charac(F) \geq
d$ from an ergodic theory counterpart, which was recently established by
Bergelson and the authors. In low characteristic we obtain a partial result, in
which the phase polynomial $\phi$ is allowed to be of some larger degree
$C(d)$. The full inverse conjecture remains open in low characteristic; the
counterexamples by Lovett-Meshulam-Samorodnitsky or Green-Tao in this setting
can be avoided by a slight reformulation of the conjecture.
