Scattered Subsets of Groups - Banakh, T. O.; Protasov, I. V.; Slobodianiuk, S. V. (Ukrains’kyi Matematychnyi Zhurnal, 2015)
We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that μ(A) = 0 for each left invariant Banach measure μ on G. It is also shown that every infinite group can be split into ℵ0 scattered subsets.