The inverse conjecture for the Gowers norm over finite fields via the correspondence principle

The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\text{F}$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\mathrm{\parallel }f{\mathrm{\parallel }}_{U^d(V)}$ if and only if it correlates with a phase polynomial $\varphi =e_{\text{F}}(P)$ of degree at most $d-1$, thus $P:V\to \text{F}$ 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 $\varphi$ 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.