Thin Subsets of Groups
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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.
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