Semiaffine sets in Abelian groups

Анотація

A subset X of an Abelian group G is called semiaffine if for every x,y,z∈X the set {x+y-z,x-y+z} intersects X. We prove that a subset X of an Abelian group G is semiaffine if and only if one of the following conditions holds: (1) X=(H+a)∪(H+b) for some subgroup H of G and some elements a,b∈ X; (2) X=(H∖C)+g for some g∈G, some subgroup H of G and some midconvex subset C of the group H. A subset C of a group H is midconvex if for every x,y∈C, the set  (x+y)/2:={z∈H:2z=x+y} is a subset of C.