  A subset of a Polish space $X$ is called universally small if it belongs to
each ccc $\sigma$-ideal with Borel base on $X$. Under CH in each uncountable
Abelian Polish group $G$ we construct a universally small subset $A_0\subset G$
such that $|A_0\cap gA_0|=\mathfrak c$ for each $g\in G$. For each cardinal
number $\kappa\in[5,\mathfrak c^+]$ the set $A_0$ contains a universally small
subset $A$ of $G$ with sharp packing index
$\pack^\sharp(A_\kappa)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$
is disjoint$\}$ equal to $\kappa$.
