Notre Dame Journal of Formal Logic

Packing Index of Subsets in Polish Groups

Taras Banakh , Nadya Lyaskovska , and Dušan Repovš


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 G such that the family of shifts x A x S is (almost) disjoint (in the sense that x A y A < G for any distinct points x , y S ). Subsets A G with small (almost) packing index are large in a geometric sense. We show that pack A 0 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 G with pack A = pack B = c and Pack ( A B ) = 1 and then apply this result to show that G contains a nowhere dense Haar null subset C G with pack(C)=Pack(C)=κ for any given cardinal number κ 4 c .

Article information

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

First available in Project Euclid: 11 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Polish group packing index Borel set Haar null set Martin axiom continuum hypothesis


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.

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