Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 4 / i

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- Solution i

Solution

With normalized averages on the finite abelian group

G

, the Gowers U2 norm is defined by

∥ f ∥_{U^{2}}^{4} = E_{x, a, b} f (x) \overline{f (x + a)} \overline{f (x + b)} f (x + a + b) .

The Gowers U3 norm is defined by

∥ f ∥_{U^{3}}^{8} = E_{x, a, b, c} \prod_{ϵ \in {0, 1}^{3}} C^{∣ ϵ ∣} f (x + ϵ_{1} a + ϵ_{2} b + ϵ_{3} c),

where

C

denotes complex conjugation.

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