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.
KeywordsAbelian Group Uniform Space Regular Cardinal Discrete Subset Limit Cardinal
Unable to display preview. Download preview PDF.
- 1.Ie. Lutsenko and I. V. Protasov, “Sparse, thin and other subsets of groups,” Int. J. Algebra Comput., 11, 491–510 (2009).Google Scholar
- 2.Ie. Lutsenko and I. V. Protasov, “Relatively thin and sparse subsets of groups,” Ukr. Math. J., 63, No. 2, 216–225 (2011).Google Scholar
- 3.Ie. Lutsenko and I. Protasov, “Thin subsets of balleans,” Appl. Gen. Topology, 11, No. 2, 89–93 (2010).Google Scholar
- 4.I. V. Protasov, “Selective survey on subset combinatorics of groups,” Ukr. Math. Bull., 7, 220–257 (2010).Google Scholar
- 8.I. Protasov and M. Zarichnyi, “General asymptology,” Math. Stud. Monogr. Ser., VNTL Publ., Lviv, 12 (2007).Google Scholar
- 9.J. Roe, “Lectures on coarse geometry,” AMS Univ. Lect. Ser., RI, Providence, 31 (2003).Google Scholar
- 10.R. Engelking, General Topology, PWN, Warszawa (1985).Google Scholar
- 14.N. Hindman and D. Strauss, Algebra in the Stone–Cěch Compactification: Theory and Applications, Walter de Gruyter, Berlin; New York (1998).Google Scholar