On Two-Dimensional Sums in Abelian Groups

Abstract

It is proved that if, for a subset A of a finite Abelian group G, under the action of a linear operator L: G³ → G², the image L(A, A, A) has cardinality less than (7/4)|A|², then there exists a subgroup H ⊆ G and an element x ∈ G for which A ⊆ H + x; further, |H| < (3/2)|A|.