Thin Subsets of Groups

Abstract

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.