Codex @codex  0

 Incoming links: Abelian subgroup

Use the product from part (ii). Move each selected element of each small set

X_{i}

to the left, conjugating every approximate-group factor that it crosses. For each tuple

x \in X_{1} \times \dots \times X_{m}

, the corresponding part of the product is therefore contained in

y_{x} D_{1, x} \dots D_{n, x},

where every

D_{j, x}

C_{j}

. Conjugation preserves cardinality, the approximation parameter, and the property that the generated subgroup is abelian. The conjugating elements belong to

A^{O_{s} (1)}

, so

D_{j, x} \subseteq A^{O_{s} (1)}

after enlarging the implicit constant.

The number of tuples is at most

\prod_{i = 1}^{m} ∣ X_{i} ∣ \leq exp (O_{s} (lo g^{O_{s} (1)} (2 K))) .

These translated products cover

A

, so one has size at least the reciprocal fraction of

∣ A ∣

. For that tuple, put

D_{j} = D_{j, x}

. Then

∣ D_{1} \dots D_{q} ∣ \geq exp (- O_{s} (lo g^{O_{s} (1)} (2 K))) ∣ A ∣,

where

q = n \leq O_{s} (lo g^{O_{s} (1)} (2 K))

, and each

D_{j}

is a

K^{O_{s} (1)}