(f_{ϵ})_{ϵ \in {0, 1}^{3}}

F_{p}^{n}

⟨ (f_{ϵ}) ⟩_{U^{3}} = E_{x, h_{1}, h_{2}, h_{3}} \prod_{ϵ \in {0, 1}^{3}} C^{∣ ϵ ∣} f_{ϵ} (x + ϵ_{1} h_{1} + ϵ_{2} h_{2} + ϵ_{3} h_{3}),

where

C

∥ f ∥_{U^{3}} = ⟨ (f)_{ϵ \in {0, 1}^{3}} ⟩_{U^{3}}^{1/8} .

∣ ⟨ (f_{ϵ}) ⟩_{U^{3}} ∣ \leq \prod_{ϵ \in {0, 1}^{3}} ∥ f_{ϵ} ∥_{U^{3}} .

Solved by gpt-5.6-sol high.

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