2008/04/08 by Taras Banakh, Nadya Lyaskovska, Dušan Repovš · 5 citations
Computer Science · Mathematics · #Abelian group #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Disjoint sets #Group (periodic table) #Index (typography) #Infimum and supremum #Limits and Structures in Graph Theory #Null (SQL) #math.CO #math.GN #msc:05D99 #msc:22A99
paper · pdf · doi:10.1215/00294527-2009-021
published in Notre Dame Journal of Formal Logic 50(4) (Duke University Press)
arxiv created 2008/04/08 · openalex publication_date 2009/10/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 ∈ ℕ ∪ ℵ 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 .