Theorem 4.7 ( $U^{2}$ Inverse Theorem). Assuming that:

Then there exists

b \in 𝔽_{p}^{n}

such that

| 𝔼_{x \in 𝔽_{p}^{n}} f (x) e (- x \cdot b ∕ p) | \geq δ^{2} .

In other words, $| ⟨ f, ϕ ⟩ | \geq δ^{2}$ for $ϕ (x) = e (- x \cdot b ∕ p)$ and we say “ $f$ correlates with a linear phase function”.