Scattered Subsets of Groups

We define the scattered subsets of a group as asymptotic counterparts of scattered subspaces of a topological space, and prove that a subset $A$ of a group $G$ is scattered if and only if $A$ contains no piecewise shifted $IP$-subsets. For an amenable group $G$ and a scattered subspace $A$ of $G$, we show that $\mu(A)=0$ for each left invariant Banach measure $\mu$ on $G$.