Semiaffine sets in Abelian groups

Abstract

A subset $X$ of an Abelian group $G$ is called $semiaf\!fine$ if for every $x,y,z\in X$ 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) $X=(H+a)\cup (H+b)$ for some subgroup $H$ of $G$ and some elements $a,b\in X$; (2) $X=(H\setminus C)+g$ for some $g\in G$, 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 $x,y\in C$, the set $\frac{x+y}2:=\{z\in H:2z=x+y\}$ is a subset of $C$.