A \subseteq G = F_{p}^{n}

α

, then

2 A - 2 A

contains a subspace of codimension at most

2 α^{- 2}

f = 1_{A}

S = {γ \in G : ∣ f (γ) ∣ \geq \frac{α ^{3/2}}{2}} .

∣ S ∣ \frac{α ^{3}}{2} \leq \sum_{γ} ∣ f (γ) ∣^{2} = α,

∣ S ∣ \leq 2 α^{- 2}

. Let

V = {x \in G : γ (x) = 1 for every γ \in S} .

Then

V

is a subspace of codimension at most

∣ S ∣

The normalized representation function of

2 A - 2 A

r (x) = (f * f * f * f) (x) = \sum_{γ \in G} ∣ f (γ) ∣^{4} γ (x),

where

f (x) = f (- x)

. For

x \in V

, all terms indexed by

S

are nonnegative real numbers, while

\sum_{γ \in / S} ∣ f (γ) ∣^{4} \leq \frac{α ^{3}}{2} \sum_{γ} ∣ f (γ) ∣^{2} = \frac{α ^{4}}{2} .

The trivial character alone contributes

α^{4}

, so

r (x) > 0

. Hence

x \in 2 A - 2 A

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