Thin Subsets of Groups - Protasov, I.V.; Slobodyanyuk, S. (Український математичний журнал, 2013)
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 |Fg ∩ A| ≤ m for all g ∈ G \ K. We show that each m-thin subset of an Abelian group G of cardinality ℵn, n = 0, 1, . . . can be split into ≤ mⁿ⁺¹ 1-thin subsets. On the other hand, we construct a group G of cardinality ℵω and select a 2-thin subset of G which cannot be split into finitely many 1-thin subsets.