Thin Subsets of Groups

For a group G and a natural number m; a subset A of G is called m-thin if, for each finite subset F of G ; there exists a finite subset K of G such that | F g ∩ A | ≤ m for all g ∈ G \ K : We show that each m -thin subset of an Abelian group G of cardinality ℵ n ; n = 0, 1,… can be split into ≤ m n +1 1-thin subsets. On the other hand, we construct a group G of cardinality ℵ ω and select a 2-thin subset of G which cannot be split into finitely many 1-thin subsets.