## Notre Dame Journal of Formal Logic

### Packing Index of Subsets in Polish Groups

#### Abstract

For a subset A of a Polish group G, we study the (almost) packing index pack( A) (respectively, Pack( 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 $|xA\cap yA|<|G|$ for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are large in a geometric sense. We show that $\pack}(A)\in\mathbb{N}\cup\{\aleph_0,\mathfrak{c}\}$ for any σ-compact subset A of a Polish group. In each nondiscrete Polish Abelian group G we construct two closed subsets $A,B\subset G$ with $\mathrm{pack}(A)=\mathrm{pack}(B)=\mathfrak{c}$ and $\mathrm{Pack}(A\cup B)=1$ and then apply this result to show that G contains a nowhere dense Haar null subset $C\subset G$ with pack(C)=Pack(C)=κ for any given cardinal number $\kappa\in[4,\mathfrak{c}]$.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 50, Number 4 (2009), 453-468.

Dates
First available in Project Euclid: 11 February 2010

https://projecteuclid.org/euclid.ndjfl/1265899125

Digital Object Identifier
doi:10.1215/00294527-2009-021

Mathematical Reviews number (MathSciNet)
MR2598874

Zentralblatt MATH identifier
1203.03066

Subjects
Primary: 03E15 03E17 03E35 03E50 03E75 05D99
Secondary: 22A99 54H05 54H11

#### Citation

Banakh , Taras; Lyaskovska , Nadya; Repovš , Dušan. Packing Index of Subsets in Polish Groups. Notre Dame J. Formal Logic 50 (2009), no. 4, 453--468. doi:10.1215/00294527-2009-021. https://projecteuclid.org/euclid.ndjfl/1265899125

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