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

Daniel El-Baz,Marc Munsch,Christoph Aistleitner

doi:10.1007/s00039-021-00564-6

Daniel El-Baz, Marc Munsch + Show 1 more

Open Access

https://doi.org/10.1007/s00039-021-00564-6

Copy DOI

Abstract

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} has been pioneered by Rudnick, Sarnak and Zaharescu. Here alpha is a real parameter, and (a_n)_{n ge 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 , in terms of the additive energy of the integer sequence (a_n)_{n ge 1}. In the present paper we develop a similar framework for the case when (a_n)_{n ge 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, the sequence (n^theta alpha )_{n ge 1} has Poissonian pair correlation for almost all alpha in {mathbb {R}}.

Highlights

Introduction and statement of resultsA sequencen≥1 of real numbers is called uniformly distributed modulo one if for all intervals A ⊂ [0, 1) the asymptotic equality lim N→∞ N 1A(yn) = λ(A) (1)n=1 holds
As an application of our method, we prove that for every real number θ > 1, the sequencen≥1 has Poissonian pair correlation for almost all α ∈ R
Uniform distribution of a sequence can be seen as a pseudorandomness property, in the sense that a sequence (Yn)n≥1 of independent, identically distributed random variables having uniform distribution on [0, 1) satisfies

Summary

Introduction and statement of results

A sequence (yn)n≥1 of real numbers is called uniformly distributed (or equidistributed) modulo one if for all intervals A ⊂ [0, 1) the asymptotic equality. Very little is known in the metric theory of pair correlation of sequences (xnα)n when (xn)n is a sequence of reals rather than integers One step in this general direction is [CLZ15], where (xn)n is allowed to take rational values and the results obtained depend on the size of the denominators of these rationals. The following theorem states that being able to control EN∗ ,γ as a function of γ allows us to deduce metric pair correlations in some cases where the condition on the additive energy in Theorem 1 fails to hold. In conclusion we note that Technau and Yesha recently obtained a result which is somewhat similar to our Theorem 3, but which is “metric” in the exponent rather than in a multiplicative parameter They showed that (nθ)n has Poissonian pair correlation for almost all θ > 7.

Preliminaries

Proof of Theorem 1

Proof of Lemma 1

Proof of Theorem 1: conclusion of the proof

Proof of Theorem 2

Proof of Theorem 3

Closing remarks