The Champernowne constant is not Poissonian

Piršić, Ísabel; Stockinger, Wolfgang

doi:10.7169/facm/1749

Cited by 8 publications

(5 citation statements)

References 19 publications

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“…Therefore, if we study the distribution of the spacings between the sequence elements of ({b n α}) n∈N , the only reasonable choice for α is a b-normal number. In [23], the sequence ({2 n α}) n∈N was studied for the Champernowne constant α and it was shown that it is not Poissonian. In this note we will choose two other special instances, which were suggested by Yann Bugeaud as potential candidates in personal communication.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Some negative results related to Poissonian pair correlation problems

Larcher

Stockinger

2020

Discrete Mathematics

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We say that a sequence (xn) n∈N in [0, 1) has Poissonian pair correlations iffor every s ≥ 0. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence (xn) n∈N of real numbers in [0, 1) having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and in more general LS-sequences of points and digital (t, 1)-sequences. Additionally, this theorem enables us to derive negative pair correlation properties for sequences of the form ({anα}) n∈N , where (an) n∈N is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of α. Second, in this note we study the pair correlation statistics for sequences of the form, xn = {b n α}, n = 1, 2, 3, . . ., with an integer b ≥ 2, where we choose α as the Stoneham number and as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property. Throughout this article {·} denotes the fractional part of a real number. arXiv:1803.05236v2 [math.NT]

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Some negative results related to Poissonian pair correlation problems

Larcher

Stockinger

2020

Discrete Mathematics

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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%

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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“…The only known positive result is (x n ) = { √ n}, see [2], where {z} := z − ⌊z⌋ denotes the fractional part of z. Indeed, many canonical candidates of uniformly distributed sequences are known to fail having Poissonian pair correlations, see [3], [5], [10]. In this paper, we work with a definition that allows us to measure how far a sequence is from having Poissonian pair correlations.…”

Section: Introductionmentioning

confidence: 99%

Sequences with almost Poissonian Pair Correlations

Weiß¹,

Skill²

2019

Preprint

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Although a generic uniformly distributed sequence has Poissonian pair correlations, only one explicit example has been found up to now. Additionally, it is even known that many classes of uniformly distributed sequences, like van der Corput sequences, Kronecker sequences and LS sequences, do not have Poissonian pair correlations. In this paper, we show that van der Corput sequences and the Kronecker sequence for the golden mean are as close to having Poissonian pair correlations as possible: they both have α-pair correlations for all 0 < α < 1 but not for α = 1 which corresponds to Poissonian pair correlations.

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The Champernowne constant is not Poissonian

Cited by 8 publications

References 19 publications

Some negative results related to Poissonian pair correlation problems

Some negative results related to Poissonian pair correlation problems

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

Sequences with almost Poissonian Pair Correlations

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