Three Dimensional Corners: A Box Norm Proof
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 G3, and d is a non-zero element of G. The Ramsey number of interest is R3(G) the maximal cardinality of a subset of G3 that does not contain a three-dimensional corner. Furstenberg and Katznelson have shown R3(ZN) is little-o of N3, 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.