Three Dimensional Corners: A Box Norm Proof

TL;DR

This paper presents a new proof for the bounds on three-dimensional corners in additive groups, extending Szemeredi's Theorem and utilizing Gowers Box Norms to unify previous results.

Contribution

It provides a novel proof of the finite field case for the maximal size of sets avoiding three-dimensional corners, generalizing Gowers and Shkredov's work.

Findings

Established that R_3(Z_N) is little-o of N^3 in finite fields.

Unified proof approach for Szemeredi's Theorem and corner results.

Extended the understanding of combinatorial structures in additive groups.

Abstract

In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a subset of G^3 that does not contain a three-dimensional corner. Furstenberg and Katznelson have shown R_3(Z_N) is little-o of N^3, and in fact the corresponding result holds in all dimensions, a result that is a far reaching extension of the Szemeredi Theorem. We give a new proof of the finite field version of this fact, a proof that is a common generalization of the Gowers proof of Szemeredi's Theorem for four term progressions, and the result of Shkredov on two-dimensional corners. The principal tool are the Gowers Box Norms.