Tile ℤ 2 with translations of one set

Abstract

Let Abe a finite subset of ℤ 2. We say Atiles 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 ∈ Cin an unique way. In this case we call Aa ℤ 2-tile and write A ⊕ C = ℤ 2. A tile Ais said to be a normal ℤ 2-tile if there exists a periodic set Csuch that A ⊕ C = ℤ 2. We characterize all normal ℤ 2-tiles with prime cardinality.