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

2007/05/03 by Shkredov, I. D.
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0705.0451

Abstract

Let G be a finite Abelian group and A be a subset G× 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.

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