Pair correlation of sequences $(\lbrace a_n α\rbrace)_{n \in \mathbb{N}}$ with maximal order of additive energy

Larcher, Gerhard; Stockinger, Wolfgang

doi:10.48550/arxiv.1802.02901

Cited by 4 publications

(8 citation statements)

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“…Recently, the result of Bourgain has been further extended, see [1,16,17,18]. The result given in [17] is an easy consequence of our Theorem 1 stated below and will be shown in Section 2.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 80%

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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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“…Recently, the result of Bourgain has been further extended, see [1,16,17,18]. The result given in [17] is an easy consequence of our Theorem 1 stated below and will be shown in Section 2.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 80%

“…Recently, this result was further improved by the authors, who showed that there is no α such that the pair correlations of ({a n α}) n∈N are Poissonian, if (a n ) n∈N is quasi-arithmetic of degree d, for d ≥ 1 ( [18]).…”

Section: Example 1: Ls-sequencesmentioning

confidence: 99%

Some negative results related to Poissonian pair correlation problems

Larcher

Stockinger

2020

Discrete Mathematics

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“…The notion Poissonian pair correlation has attracted renewed interest in the last few years, due to its connection to several mathematical fields, such as Diophantine approxmation, additive combinatorics and uniform distribution (see e.g., [1,3,4,9,11,19]). The link between the concept of uniform distribution modulo 1 and Poissonian pair correlation has been studied in the one-dimensional case.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On a multi-dimensional Poissonian pair correlation concept and uniform distribution

et al. 2019

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The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit interval at distances of order of the mean spacing 1/N . In the d-dimensional case, of course, the order of the mean spacing is 1/N 1 d , and -in our concept-the distance of sequence elements will be measured by the supremum-norm.Additionally, we show that, in some sense, almost all sequences satisfy this new concept and we examine the link to uniform distribution. The metrical pair correlation theory is investigated and it is proven that a class of typical low-discrepancy sequences in the high-dimensional unit cube do not have Poissonian pair correlations, which fits the existing results in the one-dimensional case.

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“…In this paper the authors also raised the question if an increasing sequence of distinct integers with maximal order of additive energy can have Poissonian pair correlations for almost all α. Jean Bourgain could show that the answer to this question is negative, see the appendix of [2] for details and a second problem which was also solved by Bourgain. Recently, the results of Bourgain have been further extended, see [1,9,10,11].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Champernowne constant is not Poissonian

Piršić¹,

Stockinger²

2019

Funct. Approx. Comment. Math.

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We say that a sequence (xn) n∈N in [0, 1) has Poissonian pair correlations iffor every s ≥ 0. In this note we study the pair correlation statistics for the sequence of shifts of α, xn = {2 n α}, n = 1, 2, 3, . . ., where we choose α as the Champernowne constant in base 2. Throughout this article {·} denotes the fractional part of a real number. It is well known that (xn) n∈N has Poissonian pair correlations for almost all normal numbers α (in the sense of Lebesgue), but we will show that it does not have this property for all normal numbers α, as it fails to be Poissonian for the Champernowne constant.

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Pair correlation of sequences $(\lbrace a_n α\rbrace)_{n \in \mathbb{N}}$ with maximal order of additive energy

Cited by 4 publications

References 0 publications

Some negative results related to Poissonian pair correlation problems

Some negative results related to Poissonian pair correlation problems

On a multi-dimensional Poissonian pair correlation concept and uniform distribution

The Champernowne constant is not Poissonian

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