Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 129 / 3 / c

Solution

Let

T = 1_{A} * 1_{A}

and choose

m = max {1, ⌈ lo g (1/ α)⌉} .

Apply part b with its error parameter replaced by a sufficiently small absolute multiple of

ε

V

O (m ε^{- 2}) = O (ε^{- 2} lo g (1/ α))

with the evident harmless modification when

α = 1

For

t \in V

x \in F_{p}^{n}

, and

q = 2 m

∣ (T * 1_{A}) (x + t) - (T * 1_{A}) (x) ∣ \leq a \in A \sum ∣ T (x + t - a) - T (x - a) ∣ \leq ∣ A ∣^{1 - 1/ q} ∥ τ_{t} T - T ∥_{q} .

Since

α^{- 1/ q} \leq e^{1/2}

by the choice of

m

, the bound from part b is at most

ε ∣ A ∣^{2 - 1/ q} N^{1/ q} = ε α^{- 1/ q} ∣ A ∣^{2} \leq ε ∣ A ∣^{2}

after absorbing the absolute factor into the chosen error parameter. This is the required uniform estimate for

1_{A} * 1_{A} * 1_{A}

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