Pair correlation of fractional parts derived from rational valued sequences

Chaubey, Sneha; Lanius, Melinda; Zaharescu, Alexandru

doi:10.1016/j.jnt.2014.12.014

Cited by 8 publications

(10 citation statements)

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“…Another case which has been intensively investigated is that of { αa(x) } x≥1 for {a(x)} x≥1 being a lacunary sequence, see for example [5,11,31]. For brevity we will say a sequence {a(x)} x≥1 has the metric pair correlation property if { αa(x) } x≥1 has asymptotically Poissonian pair correlations for almost all α.…”

Section: Introductionmentioning

confidence: 99%

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

2017

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Abstract. For a sequence of integers {a(x)} x≥1 we show that the distribution of the pair correlations of the fractional parts of { αa(x) } x≥1 is asymptotically Poissonian for almost all α if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of α such that { αx d } fails to have Poissonian pair correlation is at most d+2 d+3 < 1. This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least 2 d+1 .

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Section: Introductionmentioning

confidence: 99%

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

2017

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“…However, for any polynomial p ∈ Z[x] of degree at least 2, the pair correlation of (p(n)α) n is Poissonian for almost all α. For related results, see for example [6,12,36].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Very little is known in the metric theory of pair correlation of sequences (x n α) n when (x n ) n is a sequence of reals rather than integers. One step in this general direction is [12], where (x n ) n is allowed to take rational values and the results obtained depend on the size of the denominators of these rationals. A general result was obtained recently in [34], where the authors gave a criterion formulated in terms of the number of solutions of a certain Diophantine inequality.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…We will relate the counting problem in Equation (12) to the problem of bounding a twisted moment of the Riemann zeta function. Before we turn to the proof, we point out the difference between the real-number case (in this paper) and the corresponding results for the case of (x n ) n≥1 being an integer sequence.…”

Section: Proof Of Theorem 1 Part 3: Lattice Point Counting Via the Ri...mentioning

confidence: 99%

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

Aistleitner¹,

El-Baz²,

Munsch³

2020

Preprint

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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 (anα) n≥1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here α is a real parameter, and (an) 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 α, in terms of the additive energy of the integer sequence (an) n≥1 . In the present paper we develop a similar framework for the case when (an) 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 θ > 1, the sequence (n θ α) n≥1 has Poissonian pair correlation for almost all α ∈ R.

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“…A generalization from ({n d α}) n≥1 to the case of ({a(n)α}) n≥1 with a(x) ∈ Z[x] is obtained in [6]; again the pair correlations are asymptotically Poissonian for almost all α, provided that the degree of a(x) is at least 2. Another case which has been intensively investigated is that of ({a(n)α}) n≥1 for (a(n)) n≥1 being a lacunary sequence; see for example [3,11,22].…”

Section: Introductionmentioning

confidence: 99%

On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics

Aichinger

Aistleitner

Larcher

2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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The investigation of the pair correlation statistics of sequences was initially motivated by questions concerning quasi-energy-spectra of quantum systems. However, the subject has been developed far beyond its roots in mathematical physics, and many challenging number-theoretic questions on the distribution of the pair correlations of certain sequences are still open. We give a short introduction into the subject, recall some known results and open problems, and in particular explain the recently established connection between the distribution of pair correlations of sequences on the torus and certain concepts from additive combinatorics. Furthermore, we slightly improve a result recently given by Jean Bourgain in [1].

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Pair correlation of fractional parts derived from rational valued sequences

Cited by 8 publications

References 10 publications

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

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

On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics

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