Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 3 / ii

Table of contents
- Solution ii

Solution

The hypothesis says that the additive energy satisfies

E (A) \geq c ∣ A ∣^{3}

. The Balog-Szemerédi-Gowers theorem supplies

A^{'} \subseteq A

with

∣ A^{'} ∣ \geq c_{1} (c) ∣ A ∣, ∣ A^{'} + A^{'} ∣ \leq C_{1} (c) ∣ A^{'} ∣.

By the Freiman-Ruzsa theorem over a finite field,

A^{'}

lies in a subspace

H

with

∣ H ∣ \leq C_{2} (c, p) ∣ A^{'} ∣.

Thus

A^{'}

has density at least

C_{2}^{- 1}

in

H

. Applying the Finite-field Bogolyubov lemma inside

H

gives a subspace

V \subseteq 2 A^{'} - 2 A^{'} \subseteq 2 A - 2 A

of codimension bounded in terms of

c, p

alone. Therefore

∣ V ∣ \geq c^{'} (c, p) ∣ H ∣ \geq c^{'} (c, p) ∣ A ∣,

which is the additive energy produces a large subspace in a fourfold difference set result.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version