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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 gG \ 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.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 9, pp. 1245–1253, September, 2013.

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Protasov, I.V., Slobodyanyuk, S. Thin Subsets of Groups. Ukr Math J 65, 1384–1393 (2014).

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  • Abelian Group
  • Uniform Space
  • Regular Cardinal
  • Discrete Subset
  • Limit Cardinal