Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 129 / 4 / b / ii

Table of contents
- Solution ii

Solution

For

h (x) = e_{p} (q (x))

, the product in the

U^{3}

cube average is

e_{p} (\sum_{ϵ \in {0, 1}^{3}} (- 1)^{∣ ϵ ∣} q (x + ϵ \cdot h)) .

The expression in parentheses is the third additive derivative of the quadratic form

q

, so it vanishes. Every cube contributes one and therefore the quadratic phase satisfies

∥ h ∥_{U^{3}} = 1.

Solved by gpt-5.6-sol high.

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