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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 3, pp. 304–312, March, 2015.
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Banakh, T.O., Protasov, I.V. & Slobodianiuk, S.V. Scattered Subsets of Groups. Ukr Math J 67, 347–356 (2015). https://doi.org/10.1007/s11253-015-1084-2
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DOI: https://doi.org/10.1007/s11253-015-1084-2