A characterization of finite sets that 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 . We 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
On finite sets which 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 . W then write A ⊕ B = Z/MZ. Newma...
متن کاملTiling the Integers with Translates of One Finite Set
A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of nding necessary and suucient conditions for a nite set to tile the integers. For sets of prime power size, it was solved by D. Newman J. Number Theory 9 (1977), 107{111]. We solve it for sets of size having at most two prime factors. The conditions are al...
متن کاملEEH: AGGH-like public key cryptosystem over the eisenstein integers using polynomial representations
GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been new ideas that revolutionized the field. We present EEH, a GGH-Like public key cryptosystem based on the Eisenstein integers Z [ζ3] where ζ3 is a primitive...
متن کاملMultiple tilings of Z with long periods, and tiles with many-generated level semigroups
We consider multiple tilings of Z by translates of a finite multiset A of integers (called a tile). We say that a set of integers T is an A-tiling of level d if each integer can be written in exactly d ways as the sum of an element of T and an element of A. We find new exponential lower bounds on the longest period of A-tiling as a function of the diameter of A, which rejoin the exponential upp...
متن کامل3 Definability and a Nonlinear Sigma Model
The concept of definability of physics fields is introduced and a set-theoretical foundation is proposed. In this foundation, we obtain a scale invariant nonlinear sigma model and then show that quantization of the model is necessary and sufficient for definability. We also obtain com-pactification of the spatial dimensions effectively and show its equivalence to quantization. We look to provid...
متن کامل