Constructing subsets of a given packing index in Abelian groups

Abstract

By definition, the sharp packing index $\ind_P^\sharp(A)$ of a subset $A$ of an abelian group $G$ is the smallest cardinal $\kappa$ such that for any subset $B\subset G$ of size $|B|\ge \kappa$ the family $\{b+A:b\in B\}$ is not disjoint. We prove that an infinite Abelian group $G$ contains a subset $A$ with given index $\ind_P^\sharp(A)=κ$ if and only if one of the following conditions holds: (1) $2\le \kappa \le |G{|}^{+}$ and $k\notin \{3,4\}$; (2) $\kappa =3$ and $G$ is not isomorphic to ${\oplus }_{i\in I}{\mathbb{Z}}_3$; (3) $\kappa =4$ and $G$ is not isomorphic to ${\oplus }_{i\in I}{\mathbb{Z}}_2$ or to ${\mathbb{Z}}_4\oplus ({\oplus }_{i\in I}{\mathbb{Z}}_2)$.

9 pages