Sequences with almost Poissonian pair correlations

Weiß, Christian; Skill, Thomas

doi:10.1016/j.jnt.2021.07.011

Cited by 8 publications

(9 citation statements)

References 31 publications

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“…Since low-discrepancy sequences may be interpreted as sequences which are as uniformly distributed as possible, there might be examples of sequences in this class having Poissonian pair correlations. However, all attempts to find such examples have failed so far and it has even been proved for many explicit types of low-discrepancy sequences (in dimension d = 1 and also in higher dimensions) that they do not have Poissonian pair correlations, see [4,15,23,32]. In this article, we will argue why it might nonetheless be worth to keep on looking for examples of Poissonian pair correlations in the class of low-discrepancy sequences.…”

Section: Introductionmentioning

confidence: 93%

“…Using technical number theoretic arguments, the corresponding statement in d = 1 was proved for the Kronecker sequence {nϕ} and van der Corput sequences in [32]. In [29], the authors prove that Kronecker sequences ({nz}) n∈N where the partial quotients in the continued fraction expansion of z satisfy a certain growth condition possess this property, for details see Remark 2.2.…”

Section: Introductionmentioning

confidence: 98%

“…Despite this high level of uniformity, (x n ) n∈N fails to have another property of a generic uniformly distributed sequence: the Kronecker sequence of the golden mean does not have Poissonian pair correlations, [15]. On the contrary, it is known that (x n ) n∈N very tightly fails to have this property because it has α-pair correlations for all 0 < α < 1, see [32]. This paper is intended to shed more light on the connections between the properties mentioned in the first paragraph.…”

Section: Introductionmentioning

confidence: 99%

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Some connections between discrepancy, finite gap properties, and pair correlations

Weiß

2022

Monatsh Math

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A generic uniformly distributed sequence $$(x_n)_{n \in \mathbb {N}}$$ ( x n ) n ∈ N in [0, 1) possesses Poissonian pair correlations (PPC). Vice versa, it has been proven that a sequence with PPC is uniformly distributed. Grepstad and Larcher gave an explicit upper bound for the discrepancy of a sequence given that it has PPC. As a first result, we generalize here their result to the case of $$\alpha $$ α -pair correlations with $$0< \alpha < 1$$ 0 < α < 1 . Since the highest possible level of uniformity is achieved by low-discrepancy sequences it is tempting to assume that there are examples of such sequences which also have PPC. Although there are no such known examples, we prove that every low-discrepancy sequence has at least $$\alpha $$ α -pair correlations for $$0< \alpha <1$$ 0 < α < 1 . According to Larcher and Stockinger, the reason why many known classes of low-discrepancy sequences fail to have PPC is their finite gap property. In this article, we furthermore show that the discrepancy of a sequence with the finite gap property plus a condition on the distribution of the different gap lengths can be estimated. As a concrete application of this estimation, we re-prove the fact that van der Corput and Kronecker sequences are low-discrepancy sequences. Consequently, it follows from the finite gap property that these sequences have $$\alpha $$ α -pair correlations for $$0< \alpha < 1$$ 0 < α < 1 .

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Some connections between discrepancy, finite gap properties, and pair correlations

Weiß

2022

Monatsh Math

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“…The advantage of this method is that it separates other influencing factors. [23,24] CCS is a multi-dimensional coupled system, and using partial correlation analysis can avoid the influence of other factors on a single factor, making the results more reasonable. However, the focus of partial correlation analysis is too single, and using it alone can easily lead to misjudgement (Figure 4).…”

Section: Diagnosis Based On Integrated Correlation Analysismentioning

confidence: 99%

Energy savings bottleneck diagnosis of cooling system based on integrated correlation analysis

Zhu

Cui

2022

Can J Chem Eng

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The circulation cooling system is an important auxiliary system in industry and has great potential for saving energy. However, a holistic energy savings retrofit of the system can cause high costs and a low input–output ratio. To address this problem, this paper integrates grey correlation analysis and partial correlation analysis to propose a new energy savings bottleneck diagnosis method for the circulating cooling system. This method deeply analyzes the operation mechanism of the industrial circulating cooling system and summarizes the main energy‐savings factor. Through the energy savings bottleneck diagnosis method based on grey correlation analysis and partial correlation analysis, the energy savings correlation coefficient and energy savings difference coefficient are calculated, and the relationship between each energy savings factor and the specific energy consumption is accurately described and quantified. Furthermore, the system energy savings bottleneck diagnosis is performed and the quantified energy savings priority of each system link is obtained. It can effectively guide the implementation of system energy savings optimization and retrofit. Finally, the effectiveness of the diagnosis method is demonstrated by simulation verification with actual network data.

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“…In [9], it is shown that the limiting gap distribution of ({log(n)}) n∈N , where {•} denotes the fractional part of a number, has an explicit distribution which is not a Poissonian distribution but close to an exponential distribution. In [13], the limiting pair correlation function of ({log(2n − 1)/log(2)}) n∈N is explicitly calculated by exploiting the simple gap structure of this sequence. Another result in [4] describes the limiting pair correlation function of orbits of a point in hyperbolic space under the action of a discrete subgroup.…”

Section: Introductionmentioning

confidence: 99%

ON THE N-POINT CORRELATION OF VAN DER CORPUT SEQUENCES

WEIß

2023

Bull. Aust. Math. Soc.

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We derive an explicit formula for the N-point correlation $F_N(s)$ of the van der Corput sequence in base $2$ for all $N \in \mathbb {N}$ and $s \geq 0$ . The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. This constitutes the first example of an exact closed-form expression of $F_N(s)$ for all $N \in \mathbb {N}$ and all $s \geq 0$ which does not require explicit knowledge about the involved sequence. Moreover, it can be immediately read off that $\lim _{N \to \infty } F_N(s)$ exists only for $0 \leq s \leq 1/2$ .

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Sequences with almost Poissonian pair correlations

Cited by 8 publications

References 31 publications

Some connections between discrepancy, finite gap properties, and pair correlations

Some connections between discrepancy, finite gap properties, and pair correlations

Energy savings bottleneck diagnosis of cooling system based on integrated correlation analysis

ON THE N-POINT CORRELATION OF VAN DER CORPUT SEQUENCES

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