Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 2 / iii

Table of contents
- Solution iii

Solution

Choose

k

so that

∣ X ∣ \leq p^{k} < p ∣ X ∣

and take a uniformly random codimension-

k

subspace

V \leq F_{p}^{n}

. Each fixed nonzero vector lies in

V

with probability at most

p^{- k}

. The union bound gives

P ((X ∖ {0}) \cap V \neq = \emptyset) \leq (∣ X ∣ - 1) p^{- k} < 1.

Thus some

V

satisfies

V \cap X = {0}

.

For a uniformly random coset

W

of this

V

, every value

ϕ (x)

lies in

W

with probability

p^{- k}

. Therefore

E_{W} ∣ ϕ^{- 1} (W) ∣ = p^{- k} p^{n},

so some coset has inverse image

A

of density at least

p^{- k} > \frac{1}{p ∣ X ∣} \geq p^{- 1} C^{- 5} .

Part i says that

ϕ ∣_{A}

is a Freiman homomorphism, proving the large Freiman-homomorphic restriction from bounded derivative images.

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