On finite sets which tile the integers
نویسندگان
چکیده
A set of integers A is said to tile the integers if there is a set C ⊂ Z such that every integer n can be written in a unique way as n = a + c with a ∈ A and c ∈ C. Throughout this paper we will assume that A is finite. It is well known (see [7]) that any tiling of Z by a finite set A must be periodic: C = B + MZ for some finite set B ⊂ Z such that |A| |B| = M . W then write A ⊕ B = Z/MZ. Newman [7] gave a characterization of all sets A which tile the integers and such that |A| is a prime power. Coven and Meyerowitz [1] found necessary and sufficient conditions for A to tile Z if |A| has at most two prime factors. To state their result we need to introduce some notation. Without loss of generality we may assume that A,B ⊂ {0, 1, . . .} and that 0 ∈ A ∩ B. Define the characteristic polynomials
منابع مشابه
A characterization of finite sets that tile the integers
A set of integers A is said to tile the integers if there is a set C ⊂ Z such that every integer n can be written in a unique way as n = a+ c with a ∈ A and c ∈ C. Throughout this paper we will assume that A is finite. It is well known (see [7]) that any tiling of Z by a finite set A must be periodic: C = B + MZ for some finite set B ⊂ Z such that |A| |B| = M . We then write A⊕B = Z/MZ. Newman ...
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