Poissonian correlation of higher order differences

Cohen, Alex

doi:10.1016/j.jnt.2020.11.017

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…Also the statement for α-pair correlations is wellknown due to Steinerberger in [28]. An alternative proof was moreover given in [6]. It is enlightening to compare Corollary 1.3 in dimension d = 1 more closely to a result from [15] which traces back the non-Poissonian pair correlations of Kronecker sequences to their finite gap property.…”

Section: Introductionmentioning

confidence: 81%

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: 81%

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

Weiß

2022

Monatsh Math

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“…Due to a result which has been proved independently in [ALP18] and [GL17] for Poissonian pair correlations (i.e. α = 1) and which has been later generalized to weak pair correlations in [Ste18,Coh21,Wei22], we know that a sequence in [0, 1] with α-weak Poissonian pair correlations is uniformly distributed. This central property transfers to the p-adic integers as well.…”

Section: Introductionmentioning

confidence: 99%

Sequences with almost Poissonian pair correlations

Weiß

Skill

2022

Journal of Number Theory

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Poissonian correlation of higher order differences

Cited by 2 publications

References 15 publications

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

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

Sequences with almost Poissonian pair correlations

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