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We consider the action of a group G on the family P(G) of all subsets of G by the right shifts A → Ag and give the dynamical characterizations of thin, n-thin, sparse and scattered subsets. For n ∈ N, a subset A of a group G is called n-thin if g0A ∩ • • • ∩ gnA is finite for all distinct g0,. .. , gn ∈ G. Each n-thin subset of a group of cardinality ℵ0 can be partitioned into n 1-thin subsets but there is a 2-thin subset in some Abelian group of cardinality ℵ2 which cannot be partitioned into two 1-thin subsets. We eliminate the gap between ℵ0 and ℵ2 proving that each n-thin subset of an Abelian group of cardinality ℵ1 can be partitioned into n 1-thin subsets.

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ABSTRACT. The presence of disjoint dense (Borel) subsets in Tychonoff cubes, Borel suhspaces of Tychonoff cubes, and dyadic compacta is examined. Several problems are stated.KEY WORDS: resolvability of topological spaces, maximally, Borel, and Baire resolvable topological spaces, Borel, A-, Gs-, and F~-sets in Tychonoffcubes, A-, CA-, PCA-, and CPCA-sets in compact spaces, dyadic compactum, perfectly normal compact space, Baire space, dispersion character.

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