TY - JOUR

T1 - THERE IS NO KHINTCHINE THRESHOLD FOR METRIC PAIR CORRELATIONS

AU - Aistleitner, Christoph

AU - Lachmann, Thomas

AU - Technau, Niclas

N1 - Publisher Copyright:
© 2019 University College London

PY - 2019

Y1 - 2019

N2 - We consider sequences of the form (Formula presented.) mod 1, where (Formula presented.) and where (Formula presented.) is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all (Formula presented.) in the sense of Lebesgue measure, we say that (Formula presented.) has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of (Formula presented.). Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence (Formula presented.) having large additive energy which, however, maintains the metric pair correlation property.

AB - We consider sequences of the form (Formula presented.) mod 1, where (Formula presented.) and where (Formula presented.) is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all (Formula presented.) in the sense of Lebesgue measure, we say that (Formula presented.) has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of (Formula presented.). Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence (Formula presented.) having large additive energy which, however, maintains the metric pair correlation property.

KW - 05A18

KW - 11B25

KW - 11B30

KW - 11K06 (secondary)

KW - 11K60 (primary)

UR - http://www.scopus.com/inward/record.url?scp=85073069294&partnerID=8YFLogxK

U2 - 10.1112/S002557931900024X

DO - 10.1112/S002557931900024X

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AN - SCOPUS:85073069294

SN - 0025-5793

VL - 65

SP - 929

EP - 949

JO - Mathematika

JF - Mathematika

IS - 4

ER -