Pair Correlation of the Fractional Parts of $\alpha n^\theta$

TL;DR

This paper proves that for certain fractional power sequences, the pair correlation is Poissonian for all positive alpha when theta is at most 1/3, using classical Fourier analysis, addressing a longstanding open problem.

Contribution

It establishes the Poissonian pair correlation for all positive alpha when theta ≤ 1/3, for the first time for explicit alpha, using classical Fourier techniques.

Findings

Pair correlation is Poissonian for theta ≤ 1/3 and all alpha > 0.

The result applies to explicit alpha, not just almost all.

Classical Fourier analysis suffices for the proof.

Abstract

Fix $\alpha,\theta >0$, and consider the sequence $(\alpha n^{\theta} \mod 1)_{n\ge 1}$. Since the seminal work of Rudnick--Sarnak (1998), and due to the Berry--Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences have been intensively studied. In this paper we show that for $\theta \le 1/3$, and $\alpha>0$, the pair correlation function is Poissonian. While (for a given $\theta \neq 1$) this strong pseudo-randomness property has been proven for almost all values of $\alpha$, there are next-to-no instances where this has been proven for explicit $\alpha$. Our result holds for all $\alpha>0$ and relies solely on classical Fourier analytic techniques. This addresses (in the sharpest possible way) a problem posed by Aistleitner--El-Baz--Munsch (2021).