### Tile *ℤ*^{2} with translations of one set

^{2}

*Hui Rao, Yu-mei Xue*

#### Abstract

Let A be a finite subset of
ℤ

^{2}. We say A tiles ℤ^{2}with the translation set C, if any integer z∈ℤ^{2}can be represented as z_{1}+z_{2}, z_{1}∈ A, z_{2}∈ C in an unique way. In this case we call A a ℤ^{2}-tile and write A ⊕ C = ℤ^{2}. A tile A is said to be a normal ℤ^{2}-tile if there exists a periodic set C such that A ⊕ C = ℤ^{2}. We characterize all normal ℤ^{2}-tiles with prime cardinality.Full Text: PDF PostScript