Definition 4.9 ( $U^{3}$ inner product). Given functions $f_{𝜀} : G \to ℂ$ for $𝜀 \in {0, 1}^{3}$ , define their $U^{3}$ inner product by

{⟨ {(f_{𝜀})}_{𝜀 \in {0, 1}^{3}} ⟩}_{U^{3} (G)} = 𝔼_{x, h_{1}, h_{2}, h_{3} \in G} \prod_{𝜀 \in {0, 1}^{3}} C^{| 𝜀 |} f_{𝜀} (x + 𝜀 \cdot h) .