On thin-complete ideals of subsets of groups

Given a family $F$ of subsets of a group $G$ we describe the structure of its thin-completion $\tau^*(F)$, which is the smallest thin-complete family that contains $I$. A family $F$ of subsets of $G$ is called thin-complete if each $F$-thin subset of $G$ belongs to $F$. A subset $A$ of $G$ is called $F$-thin if for any distinct points $x,y$ of $G$ the intersection $xA\cap yA$ belongs to the family $F$. We prove that the thin-completion of an ideal in an ideal. If $G$ is a countable non-torsion group, then the thin-completion $\tau^*(F_G)$ of the ideal $F_G$ of finite subsets of $G$ is coanalytic but not Borel in the power-set $P_G$ of $G$.Comment: 10 page

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]

Asymptotic dimension and small subsets in locally compact topological groups

We prove that for a coarse space $X$ the ideal $S(X)$ of small subsets of $X$ coincides with the ideal $D_<(X)$ of subsets $A\subset X$ of asymptotic dimension $asdim(A)<asdim(X)$ provided that $X$ is coarsely equivalent to an Euclidean space $R^n$. Also we prove that for a locally compact Abelian group $X$, the equality $S(X)=D_<(X)$ holds if and only if the group $X$ is compactly generated.Comment: 10 page