Theorem 4.14 ( $U^{3}$ inverse theorem). Assuming that:

$f : 𝔽_{5}^{n} \to ℂ$
$∥ f ∥_{L^{\infty} (𝔽_{5}^{n})} \leq 1$
$∥ f ∥_{U^{3} (G)} \geq δ$ for some $δ > 0$

Then there exists a symmetric

n \times n

matrix

M

with entries in

𝔽_{5}

and

b \in 𝔽_{5}^{n}

such that

| 𝔼_{x} f (x) e ((x^{⊤} M x + b^{⊤} x) ∕ p) | \geq c (δ)

where $c (δ)$ is a polynomial in $δ$ . In other words, $| ⟨ f, ϕ ⟩ | \geq c (δ)$ for $ϕ (x) = e ((x^{⊤} M x + b^{⊤} x) ∕ p)$ and we say “ $f$ correlates with a quadratic phase function”.