 Quadratic phase detection by the Gowers U2 norm

ID: quadratic-phase-detection-by-the-gowers-u2-norm

Let

∣ f ∣ \leq 1

Z_{N}

and put

g (x) = f (x) ω^{- x^{2}}

. If

∥ g ∥_{U^{2}}^{4} \geq c

c \leq \sum_{r} ∣ g (r) ∣^{4} \leq (max_{r} ∣ g (r) ∣^{2}) \sum_{r} ∣ g (r) ∣^{2} \leq max_{r} ∣ g (r) ∣^{2} .

Consequently

f

correlates with a quadratic phase: for some

r \in Z_{N}

E_{x} f (x) ω^{- r x - x^{2}} \geq c^{1/2} .

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