Covering and packing in {\mathbb Z^n} and {\mathbb R^n}. (II)

In the present part (II) we will deal with the group \mathbb G = \mathbb Zⁿ{\mathbb G = \mathbb Z^n} , and we will study the effect of linear transformations on minimal covering and maximal packing densities of finite sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} . As a consequence, we will be able to show that the set of all densities for sets A{\mathcal A} of given cardinality is closed, and to characterize four-element sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} which are “tiles”. The present work will be largely independent of the first part (I) presented in 4].