Addition theorems for sets of integers

Long, Calvin T.

doi:10.2140/pjm.1967.23.107

Cited by 38 publications

(27 citation statements)

References 4 publications

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“…The paper [Lon67] contains some of what we were doing in part of this section. We are including details here nonetheless for the benefit of the reader, and in order to stress what is needed for our purpose.…”

Section: Unions Of Intervals As Affine Ifssmentioning

confidence: 99%

“…. , n − 1} were completely classified in [Lon67]. The classification is based on the next two Lemmas.…”

Section: Unions Of Intervals As Affine Ifssmentioning

confidence: 99%

“…Especially the examples in section 4 owe much to what we learned from Bent Fuglede and Steen Pedersen. The authors thank the referee for insightful suggestions regarding the presentation, and for adding to the list of references, especially [IKT01] and [Lon67].…”

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confidence: 99%

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Quasiperiodic spectra and orthogonality for iterated function system measures

Dutkay

Jørgensen

2008

Math. Z.

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Abstract. We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a "small perturbation" of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices.

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Section: Unions Of Intervals As Affine Ifssmentioning

confidence: 99%

“…. , n − 1} were completely classified in [Lon67]. The classification is based on the next two Lemmas.…”

Section: Unions Of Intervals As Affine Ifssmentioning

confidence: 99%

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Quasiperiodic spectra and orthogonality for iterated function system measures

Dutkay

Jørgensen

2008

Math. Z.

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“…If A+BN then we say that B is a complementing set for A in N. Clearly, this is a symmetric property in A and B. De Bruijn [1] studied the behavior of complementing subsets of N, and the structure of such subsets is well understood, see [1] and [7]. When the set N is replaced by the set Z, the set of all integers, then the situation gets to be much more complicated, and very little seems to be known about the behavior of complementing subsets of Z.…”

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confidence: 99%

Complementing sets of integers-A result from ergodic theory

1992

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Weakly wandering sequences play an important role in the study of ergodic measure preserving transformations defined on an infinite measure space. A se quence {ni} is a weakly wandering sequence for a transformation T defined on a measure space (X, B, Vie) if there exists a set A of positive measure for which the sets Tni A are mutually disjoint. In the past we have been successful in utilizing these sequences in exhibiting new and useful properties of ergodic transformations. In this article we discuss an application of weakly wandering sequences to the additive behavior of integers. We present here a result and a proof that are number theoretic in nature. We should mention however, that the measure preserving transforma tions that we have been studying together with their ergodic properties are the underlying concrete structures from which we were able to conjecture and prove the algebraic results that follow. In this article we omit discussion of topics that are ergodic in nature. To the interested reader we refer the articles [2] through [6]. Let us consider the set of all non-negative integers N={0, 1, 2,...}. Two infinite subsets A and B of N are said to have a direct sum if whenever an integer nEN can be represented as n=a+b and n=a'+b' with a, a'EA and b, b'EB then a=a' and b=b'. In this case we use the notation A+B to indicate the set {a+baEA, beB} . If A+BN then we say that B is a complementing set for A in N. Clearly, this is a symmetric property in A and B. De Bruijn [1] studied the behavior of complementing subsets of N, and the structure of such subsets is well understood, see [1] and [7]. When the set N is replaced by the set Z, the set of all integers, then the situation gets to be much more complicated, and very little seems to be known about the behavior of complementing subsets of Z. For instance, if an infinite set A has a complementing set B in N then this complementing set in N is unique. It is also not difficult to show that the set A has the set -B={-bbEB} as a complementing set in Z, and in fact, the set A happens to possess a continuum number of complementing subsets in Z; see [3]. A typical example in this connection is the set A consisting of 0 and all finite sums of odd powers of 2. One may regard the set A as consisting of 0 and all positive integers n which contain zero in the even places of their binary representation.It is easy to see that A has as its complementing set in N the set B consisting

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“…The result, which was implicit from the work of de Bruijn [2], was first explicitly by Long [5]. It is the following: In case \A\<oo or | J?…”

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confidence: 99%