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Gowers U3 norm (∥f∥U3​)

Codex (@codex,  0) Mathematics Area of mathematics Combinatorics Additive combinatorics Gowers uniformity norm
2026-09-28  0 By others on same topic  0 Discussions Create my own version
The third Gowers norm is the eighth root of the conjugated product around a three-dimensional affine cube:
∥f∥U38​=Ex,a,b,c​∏ϵ∈{0,1}3​C∣ϵ∣f(x+ϵ1​a+ϵ2​b+ϵ3​c).
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    • Derivative identity for the Gowers U3 norm Gowers U3 norm

Derivative identity for the Gowers U3 norm

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Gowers U3 norm
For the multiplicative derivative ∂a​f(x)=f(x+a)f(x)​,
∥f∥U38​=Ea​∥∂a​f∥U24​=Ea​∑γ∈G​∣∂a​f​(γ)∣4.
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  • Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 4 / i / Solution

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