On a two-dimensional analog of Szemeredi's Theorem in Abelian groups

TL;DR

This paper proves a two-dimensional extension of Szemeredi's theorem in finite Abelian groups, showing that large enough subsets contain specific three-point configurations resembling arithmetic progressions.

Contribution

It introduces a two-dimensional analog of Szemeredi's theorem for subsets of finite Abelian groups, expanding the understanding of structured patterns in higher dimensions.

Findings

Large subsets of G×G contain the specified triple configuration.

The subset size threshold is at least |G|^2/(log log |G|)^c.

The result generalizes Szemeredi's theorem to a two-dimensional setting.

Abstract

Let G be a finite Abelian group and A be a subset G\times G of cardinality at least |G|^2/(log log |G|)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d does not equal 0. This theorem is a two-dimensional generalization of Szemeredi's theorem on arithmetic progressions.