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 $\kappa\in[4,\mathfrak{c}]$ .
Publié le : 2009-10-15
Classification:
Polish group,
packing index,
Borel set,
Haar null set,
Martin axiom,
continuum hypothesis,
03E15,
03E17,
03E35,
03E50,
03E75,
05D99,
22A99,
54H05,
54H11
@article{1265899125,
author = {Banakh , Taras and Lyaskovska , Nadya and Repov\v s , Du\v san},
title = {Packing Index of Subsets in Polish Groups},
journal = {Notre Dame J. Formal Logic},
volume = {50},
number = {1},
year = {2009},
pages = { 453-468},
language = {en},
url = {http://dml.mathdoc.fr/item/1265899125}
}
Banakh , Taras; Lyaskovska , Nadya; Repovš , Dušan. Packing Index of Subsets in Polish Groups. Notre Dame J. Formal Logic, Tome 50 (2009) no. 1, pp. 453-468. http://gdmltest.u-ga.fr/item/1265899125/