Tiling the Integers with Translates of One Finite Set
نویسندگان
چکیده
منابع مشابه
Tiling 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...
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Definition 1.1 is motivated by a conjecture of Fuglede [2], which asserts that a measurable set E ⊂ Rn tiles Rn by translations if and only if the space L2(E) has an orthogonal basis consisting of exponential functions {e}λ∈Λ; the set Λ is called a spectrum for E. For recent work on Fuglede’s conjecture see e.g. [4], [5], [6], [7], [8], [9], [11], [12], [13], [14], [15], [16], [18], [20], [21],...
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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 z1 + z2, z1 ∈ A, z2 ∈ C in a 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.
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A finite subset A of integers tiles the discrete line Z if the integers can be written as a disjoint union of translates of A. In some cases, necessary and sufficient conditions for A to tile the integers are known. We extend this result to a large class of nonperiodic tilings and give a new formulation of the Coven-Meyerowitz reciprocity conjecture which is equivalent to the Flugede conjecture...
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Let E ⊆ Fq be a set in the 2-dimensional vector space over a finite field with q elements, which satisfies |E| > q. There exist x, y ∈ E such that |E · (y− x)| > q/2. In particular, (E + E) · (E− E) = Fq.
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 1999
ISSN: 0021-8693
DOI: 10.1006/jabr.1998.7628