Pair correlation of sequences  with maximal additive energy

Larcher, Gerhard; Stockinger, Wolfgang

doi:10.1017/s030500411800066x

Cited by 15 publications

(16 citation statements)

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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%

“…Theorem 4 There exists an absolute positive constant C such that the following is true. Let A N denote the first N elements of (a n ) n∈N and suppose that However, if the additive energy is of maximal order, i.e., if we have E(A N ) = Ω(N 3 ), then there is no α such that ({a n α}) n∈N has Poissonian pair correlations, see [11]. The approach used in [11] can be generalised to arbitrary dimensions.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…To prove the result in the one-dimensional case, we mention that most of the arguments are based on the additive structure of the integer sequence (a n ) n∈N , i.e., the claim that there are "many" difference vectors u for which we have "many" pairs (k, l) such that u = r k − r l (cf., Property 1 in the proof of the main result of [11]) does not change in the multi-dimensional setting. For Property 2 in [11], one has to consider now an interval of the form β, β + L γN 1/d d , for some constants β, γ, L > 0. The remaining arguments following Property 2, however, do not change either and also note that in our definition of the supremum-norm, we work with the distance modulo 1 instead of the absolute value.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…As the proof uses exactly the same steps, except some minor technical changes, as the one in the one-dimensional case, see [11], we will omit a detailed illustration. To prove the result in the one-dimensional case, we mention that most of the arguments are based on the additive structure of the integer sequence (a n ) n∈N , i.e., the claim that there are "many" difference vectors u for which we have "many" pairs (k, l) such that u = r k − r l (cf., Property 1 in the proof of the main result of [11]) does not change in the multi-dimensional setting. For Property 2 in [11], one has to consider now an interval of the form β, β + L γN 1/d d , for some constants β, γ, L > 0.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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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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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

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

et al. 2019

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“…Pirsic and Stockinger [9] proved it for Champernowne's constant (defined in base b). Larcher and Stockinger [6] proved it for x a Stoneham number [12] and for every real number x having an expansion which is an infinite de Bruijn word (see [3,13] for the presentation of these infinite words). Larcher and Stockinger in [7] also show the failure of the property for other sequences of the form (a n x mod 1) n≥1 .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Low discrepancy sequences failing Poissonian pair correlations

2019

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M. Levin defined a real number x that satisfies that the sequence of the fractional parts of (2 n x) n≥1 are such that the first N terms have discrepancy O((log N ) 2 /N ), which is the smallest discrepancy known for this kind of parametric sequences. In this work we show that the fractional parts of the sequence (2 n x) n≥1 fail to have Poissonian pair correlations. Moreover, we show that all the real numbers x that are variants of Levin's number using Pascal triangle matrices are such that the fractional parts of the sequence (2 n x) n≥1 fail to have Poissonian pair correlations.

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A pair correlation problem, and counting lattice points with the zeta function

2021

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The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $$(a_n \alpha )_{n \ge 1}$$ ( a n α ) n ≥ 1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here $$\alpha $$ α is a real parameter, and $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $$\alpha $$ α , in terms of the additive energy of the integer sequence $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 . In the present paper we develop a similar framework for the case when $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number $$\theta >1$$ θ > 1 , the sequence $$(n^\theta \alpha )_{n \ge 1}$$ ( n θ α ) n ≥ 1 has Poissonian pair correlation for almost all $$\alpha \in {\mathbb {R}}$$ α ∈ R .

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Pair correlation of sequences with maximal additive energy

Abstract: We show for sequences (an) n∈N of distinct positive integers with maximal order of additive energy, that the sequence ({anα}) n∈N does not have Poissonian pair correlations for any α. This result essentially sharpens a result obtained by J. Bourgain on this topic.

Cited by 15 publications

References 20 publications

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

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

Low discrepancy sequences failing Poissonian pair correlations

A pair correlation problem, and counting lattice points with the zeta function

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