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

Abstract

Fix $\alpha ,\theta >0$, and consider the sequence $(\alpha n^{\theta }\mathrm{mod}1{)}_{n\ge 1}$. 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 $\theta \ne 1$) 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).

Revised version, accepted for publication in Journal of European Mathematical Society