Corners theorem

In arithmetic combinatorics, the corners theorem states that for every

ε > 0

contains a corner, i.e., a triple of points of the form

It was first proved by Miklós Ajtai and Endre Szemerédi in 1974 using Szemerédi's theorem.

[1] In 2003, József Solymosi gave a short proof using the triangle removal lemma.

[2] Define a corner to be a subset of

, there exists a positive integer

is dense, then it has some dense subset that is centrally symmetric.

What follows is a sketch of Solymosi's argument.

Construct an auxiliary tripartite graph

corresponds to the line

Connect two vertices if the intersection of their corresponding lines lies in

Note that a triangle in

corresponds to a corner in

, except in the trivial case where the lines corresponding to the vertices of the triangle concur at a point in

is in exactly one triangle, so by the triangle removal lemma,

be the size of the largest subset of

The lower bound is due to Green,[3] building on the work of Linial and Shraibman.

[4] The upper bound is due to Shkredov.

is a set of points of the form

is the standard basis of

The natural extension of the corners theorem to this setting can be shown using the hypergraph removal lemma, in the spirit of Solymosi's proof.

The hypergraph removal lemma was shown independently by Gowers[6] and Nagle, Rödl, Schacht and Skokan.

[7] The multidimensional Szemerédi theorem states that for any fixed finite subset

, there exists a positive integer

contains a subset of the form

This theorem follows from the multidimensional corners theorem by a simple projection argument.

[6] In particular, Roth's theorem on arithmetic progressions follows directly from the ordinary corners theorem.