2019
Том 71
№ 8

All Issues

Slobodianiuk S. V.

Articles: 3
Article (English)

Ultrafilters on balleans

Protasov I. V., Slobodianiuk S. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1698-1706

A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter satellites of a ballean (namely, corona, companion, and corona companion), evaluate the basic cardinal invariants of the corona and characterize the subsets of balleans in terms of companions.

Article (English)

Scattered Subsets of Groups

Banakh T. O., Protasov I. V., Slobodianiuk S. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 304-312

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that μ(A) = 0 for each left invariant Banach measure μ on G. It is also shown that every infinite group can be split into ℵ0 scattered subsets.

Article (English)

Thin Subsets of Groups

Protasov I. V., Slobodianiuk S. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1245–1253

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.