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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ie. Lutsenko and I. V. Protasov, “Sparse, thin and other subsets of groups,” Int. J. Algebra Comput., 11, 491–510 (2009).
Ie. Lutsenko and I. V. Protasov, “Relatively thin and sparse subsets of groups,” Ukr. Math. J., 63, No. 2, 216–225 (2011).
Ie. Lutsenko and I. Protasov, “Thin subsets of balleans,” Appl. Gen. Topology, 11, No. 2, 89–93 (2010).
I. V. Protasov, “Selective survey on subset combinatorics of groups,” Ukr. Math. Bull., 7, 220–257 (2010).
I. Protasov, “Partitions of groups into thin subsets,” Algebra Discrete Math., 11, 88–92 (2011).
T. Banakh and N. Lyaskovska, “On thin complete ideals of subsets of groups,” Ukr. Math. J., 63, No. 6, 741–754 (2011).
O. Petrenko and I. V. Protasov, “Thin ultrafilters,” Notre Dame J. Formal Logic., 53, 79–88 (2012).
I. Protasov and M. Zarichnyi, “General asymptology,” Math. Stud. Monogr. Ser., VNTL Publ., Lviv, 12 (2007).
J. Roe, “Lectures on coarse geometry,” AMS Univ. Lect. Ser., RI, Providence, 31 (2003).
R. Engelking, General Topology, PWN, Warszawa (1985).
J. C. Simms, “Another characterization of Alephs: decompositions of hyperspace,” Notre Dame J. Formal Logic., 38, 19–36 (1997).
R. O. Davies, “The power of the continuum on some propositions of the plane geometry,” Fundam. Math., 52, 277–281 (1963).
R. O. Davies, “On a problem of Erd¨os concerning decomposition of the plane,” Proc. Cambridge Phil. Soc., 59, 33–36 (1963).
N. Hindman and D. Strauss, Algebra in the Stone–Cěch Compactification: Theory and Applications, Walter de Gruyter, Berlin; New York (1998).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 9, pp. 1245–1253, September, 2013.
Rights and permissions
About this article
Cite this article
Protasov, I.V., Slobodyanyuk, S. Thin Subsets of Groups. Ukr Math J 65, 1384–1393 (2014). https://doi.org/10.1007/s11253-014-0866-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0866-2