Author:
Aistleitner Christoph,El-Baz Daniel,Munsch Marc
Abstract
AbstractThe 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
.
Funder
Graz University of Technology
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Analysis
Cited by
6 articles.
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