Packing index of subsets in Polish groups

For a subset $A$ of a Polish group $G$, we study the (almost) packing index \ind_P(A) (resp. \Ind_P(A)) of $A$, equal to the supremum of cardinalities $|S|$ of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$ is (almost) disjoint (in the sense that $|xA\cap yA|<|A|$ for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are small in a geometric sense. We show that \ind_P(A)\in \IN\cup\{\aleph_0,\cc\} for any $\sigma$-compact subset $A$ of a Polish group. If $A\subset G$ is Borel, then the packing indices \ind_P(A) and \Ind_P(A) cannot take values in the half-interval [\sq(\Pi^1_1),\cc) where \sq(\Pi^1_1) is a certain uncountable cardinal that is smaller than \cc in some models of ZFC. In each non-discrete Polish Abelian group $G$ we construct two closed subsets $A,B\subset G$ with \ind_P(A)=\ind_P(B)=\cc and \Ind_P(A\cup B)=1 and then apply this result to show that $G$ contains a nowhere dense Haar null subset $C\subset G$ with \ind_P(C)=\Ind_P(C)=\kappa for any given cardinal number \kappa\in[4,\cc]

Decomposing the real line into Borel sets closed under addition

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in $ZFC$ and even in the theory $ZFC + \mathfrak{c} = \omega_2$ if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition $(0,\infty)$, and so on

Extremal densities and measures on groups and GG-spaces and their combinatorial applications

This text contains lecture notes of the course taught to Ph.D. students of Jagiellonian University in Krakow on 25-28 November, 2013.Comment: 18 page

Completeness of invariant ideals in groups

We introduce and study various notions of completeness of translation-invariant ideals in groups.Введено та досліджено різні поняття повноти інваріантних ідеалів у групах

Prethick subsets in partitions of groups

A subset S of a group G is called thick if, for any finite subset F of G, there exists g ∈ G such that Fg ⊆ S, and k-prethick, k ∈ N if there exists a subset K of G such that |K| = k and KS is thick. For every finite partition P of G, at least one cell of P is k-prethick for some k ∈ N. We show that if an infinite group G is either Abelian, or countable locally finite, or countable residually finite then, for each k ∈ N, G can be partitioned in two not k-prethick subsets

Densities, submeasures and partitions of groups

In 1995 in Kourovka notebook the second author asked the following problem: it is true that for each partition $G=A_1\cup\dots\cup A_n$ of a group $G$ there is a cell $A_i$ of the partition such that $G=FA_iA_i^{-1}$ for some set $F\subset G$ of cardinality $|F|\le n$? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups.Comment: 14 pages (this is an update of the preceding version