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

Abstract

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 $\Vert f{\Vert }_{U^d(V)}$ if and only if it correlates with a phase polynomial $ϕ= e_\F(P)$ 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 $\varphi$ 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.

21 pages, no figures, to appear, Analysis & PDE. This is the final version, incorporating the referee's corrections

References in corpus (1)

• New bounds for Szemeredi's Theorem, I: Progressions of length 4 in finite field geometries

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