Pair correlation of the fractional parts of α nθ

Fix α,θ >0 and consider the sequence (α nθ\; mod\; 1)n≥ 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 monomial sequences have been intensively studied. In this paper, we show that for θ <14/41 and α>0 , the pair correlation function is Poissonian. While (for a given θ ≠ 1 ) this strong pseudo-randomness property has been proven for almost all values of α , there are next-to-no instances where this has been proven for explicit α . Our result holds for all α>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).