Scattered Subsets of Groups
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.
English version (Springer): Ukrainian Mathematical Journal 67 (2015), no. 3, pp 347-356.
Citation Example: Banakh T. O., Protasov I. V., Slobodianiuk S. V. Scattered Subsets of Groups // Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 304-312.